十 - 広島大学学術情報リポジトリ - Hiroshima University

Transcription

十「十;.':'
International Cooperation Project
Towards the Endogenous Development of Mathematics Education
開発途上国における理数科教育協力の評価指標
に関する実証的研究
一農村部児童の基礎学力の充実を中心にEmpirical Study on the Evaluation Method
for International Cooperation in Mathematics Education
in Developing Countries
° °
-
Focusing
on
Pupils'Learning
Achievement一
平成16-18年度科学研究費補助金
(課題番号16402046)
2004-2006 Scientific Research Fund
No. 16402046
研究成果報告書
Final Report
平成19年3月
2007 March
研究代表者:岩崎秀樹
広島大学大学院教育学研究科教授
Research Leader Hideki IWASAKI
Professor, Graduate School of Education
Hiroshima University
0100A53478
Table of Contents
Preface
Chapter
1
Research
Iiackground
and
Methodology
v
----・・・・….…..…..….…
1
1. 1 Research background and objective
1
1.2 Research members
2
1.3 Contents of report
3
1.4 Research methodology
4
1.5 Plan of activity
8
Chapter
2 Country
Report
ll
2. 1 Bangladesh
ll
2.2 China
25
2.3 Ghana
36
2.4 Philippines
51
2.5 Thailand
66
2.6 Zambia
82
2.7 Japan
Chapter 3 Discussions and Future Issues 117
3. 1 Discussions of present status of mathematics education
3.2 Future issues
Postface 123
Special Contribution.….…‥………….…………………..……‥…..…‥….…… 125
1. Numeracy and mathematics - a cultural perspective on the relationship ------ 125
2. The world role of culture in mathematics education
3.
Teaching
for
numeracy:
Re-examining
the
role
of
cultural
aspects
of
mathematics
…
160
4. Minority students and teacher's support: reviewing international assessment results
and cultural approach
Literature Review..…..…………….….……...…….………….…..‥……….…. 177
Prior study of the formation of figural concepts
ANNEX: Research Tools ….… ….……… … … …..183
1. Questionnaire for the fist year survey
2. Achievement tests for the first year survey
3. Achievement test for the second year survey
4. Interview items for the second year survey
iii
Preface
International Cooperation Project Towards the Endogenous Development of Mathematics Education
Final Report
まえがき
基礎教育の完全普及を目指す「万人のための教育(以下、 EFA : Education for All)」の理
念は、 1990年タイのジョムティエンで一堂に会した国際機関、先進国、開発途上国に
共有されている教育開発目標であった。しかし、 2000年にセネガル共和国の首都ダカ
ールで開催された世界教育フォーラムでは、途上国の教育を取り巻く様々な問題が未解
決のままであることが指摘された。行動目標型で評価を語る試みは、様々な形でかつ
様々な国際協力機関でなされているが、ありうべき評価の枠組みを確立するためには継
続的かつ基礎的な研究が必要不可欠である。国際理数科教育協力においても「授業」に注
目する意義が語られる一方で、我々は途上国の授業の実相とその歴史的・社会文化的背
景を真に把握しているとはいいがたい。
教育協力における評価の枠組みは、各国の教育経験に大きく影響される。換言すれば、
評価指標は対処療法的である。現行の国際教育協力においても授業研究は重要な位置を
占めているものの、教育開発の一手段的な位置づけであり、プロジェクト評価全体の枠
組みはいまだ事前と事後の比較による行動主義的な評価という傾向が強い。行動主義的
評価における明確さ、客観化、システム化、数量化、効率化を特徴とする方法論の重要
性を軽視するものではないが、そうした評価枠組みだけでは国際教育協力にとって本質
なこと(たとえば、組織の内発性、自立発展性、教師の自律性、職業意識、子どもの創
造的な思考力、生きる力など)を明確にできない。従って、評価手法を開発する際に「測
れないもの」の存在を意識し、手法の有効性と同時に限界を十分に踏まえて多面的な視
座から開発にあたることが重要である。複雑で重層的な授業を対象にして、それを適切
に評価分析するためには、事前一事後の行動主義的評価が-項目となるような、新たな
評価枠組みの開発が急務といえよう。
本研究の目的は、行動主義的な「測定」と「予測」よりはむしろ「記述」と「説明」を重視し、
複数の国を対象としてプロセス重視の評価項目と手法を確立することであった。途上国
の特に算数における教授一学習環境や児童の基礎学力の実際を把握する際に、これらに
影響を及ぼす教育の歴史性、社会文化性に注目したことは必然であったといえよう。
本研究では、異なる歴史的、社会文化的背景をもつ途上国の具体的な教育情報は不可
欠であり、平成16年度と平成17年度に現地調査を実施した。また、次のように毎年ワ
ークショップを日本で開催し、課題の具体化とその解決に向けた議論を共有し蓄積した。
平成16年度数学教育の内発的発展に向けた第1回国際ワークショップ
日時:平成17年3月15日より17日まで
場所:広島大学大学院国際協力研究科
平成一17年度数学教育の内発的発展に向けた第2回国際ワークショップ
日時:平成17年12月11日より13日まで
場所:広島大学大学院国際協力研究科
平成18年度国際教育協力シンポジウム2007 「数学教育の自立的発展に向けた国際協
力のあり方」
日時:平成19年1月14日
場所:国際協力機構(JICA)国際協力総合研修所
V
Final Report
まず、この3年間をとおして、オーストラリア国モナシュ大学名誉教授のAlanJ.
Bishop先生の助言を仰ぐことができたことに感謝したい。また、第3回研究集会は、 3
年間の研究成果を確認する意味で、筑波大学教育開発国際協力センター(CRICED)の支
援を得て、文部科学省・国際協力機構(JICA)日本数学教育学会・日本科学教育学会の
後援の下、東京で開催することができた。教育開発に関心をお持ちの多くの参加者に向
けて、米国イリノイ州立大学のNormaC.Presmeg先生による「数学教育における文化の
役割」と題する基調講演を企画できたことは、本科研の持尾を飾るにはできすぎた展開
であった。本報告書は第3回国際ワークショップで発表された論文をもとにまとめたも
のである。したがってワークショップの詳細については、本文をお読みいただきたい。
最後になったが本書をこのような形に編集できたのも、また3度にわたるワークショ
ップを何とか実施できたのも、研究分担者・協力者の方々の尽力に負うことがほとんど
であった。ここにその名や研究発表実績を記して謝意を表しておきたい。
【国内研究組織】
氏名
研究代表者
岩崎秀樹
研究分担者
植田敦三
馬場卓也
磯田正美
丸山英樹
桑山尚司
研究協力者
二宮裕之
小原豊
中村聡
U d din M D M oh sm
内田豊海
木根主税
L ev iE h pan e
所属 . 部局 .職名
広島大学 ..大学院教育学研究科 . 教授
広島大学 . 大学院教育学研究科 . 教授
広島大学 . 大学院国際協力研究科 . 助教授
筑波大学 . 教育開発国際協力センター . 助教授
国立教育政策研究所 . 国際研究協力部 . 研究員
広島大学 . 大学院教育学研究科 . 助手
埼玉大学 . 教育学部 . 助教授
鳴門教育大学 . 教員教育国際協力センター . 助教授
広島大学 . 大学院国際協力研究科 . 院生
埼玉大学 .教育学部 .院生
【国外研究組織】
氏名
所属 .部局 .職名
研究協力者
A lan J.B ish op
W ee T io ng S eah
A .H .M .M o h iud d in
D ai(〕u in
Jin K ang b iao
P ro fessor E m eritu s, M o nash U n iv ersity,A u stralia
L ecturer, P rofessor E m eritus, M o nash U n iv ersity,A u stralia
Sp ecialist, N A P E (T he N atio nal A cadem y fo r P rim ary E du cation ),
B an g ladesh
P ro fessor, In ner M o ng olia U n iversity of E du cation , C h in a
G rad uate of ID E C ,C h in a
Vl
Final Report
Er
n estD avis K of
i
G hartey A m piah
N arh Francis
L ecturer,U niversity of C ape C oast,G hana
Senior Lecturer,U niversity of C ape C oast,G hana
M ilagrosD .Ibe
L ecturer,M ountM ary C ollege of E ducation,G hana
E m eritus Professor, D ean, U niversity of the Philippines-D ilim an,
the Philippines
M aitree Inprasitha
N orm a Presm eg
B entry N khata
A ssociate Professor,K hon K aen U niversity,T hailand
Professor,Illinois State U niversity,U SA
L ecturer,U niversity of Zam bia,Z am bia
【研究発表】
(1)学会誌等
Baba,T., Iwasaki,H. "Intersection of Critical Mathematics Education and Ethnic mathematics",
Journal ofInner Mongolia Normal University 1 9, 2006, pp.77-84.
Maruyama,
H.,バDynamics
around
Educational
Research
towards
International
Development
Cooperation - Research Approach for All Lifelong Learners", Sasai, H. & Maruyama, H. (Ed.),
Development Cooperation and Academism in Non-Formal Education - Sweden and Germany
as case studies, Study Group of International Cooperation in Non-Formal Education,
Asia仲acific Cultural Centre for UNESCO, 2005, pp.7 1 -83.
Mohsin, U. MD, Baba, T., "Analysis of Primary Mathematics in Bangladesh from Pupils'and
Teachers' Perspectives - Focusing on Fraction", International Journal of Curriculu崩
Development and Practice, 9(1 ), 2007, pp.37-53.
Praktipong, N, Nakamura, S., "Analysis of Mathematics Performance of Grade Five Students in
Thailand Using Newman Procedure", Journal of International Cooperation in Education,9{¥ ),
2006, pp.111-122
(2)学会発表
内田豊海、中村聡、馬場卓也「ザンビア基礎学校における生徒の分数概念獲得の到達段階
について」国際開発学会第17回全国大会、 2006年11月24・25日、東京大学.
【交付決定額(配分額)】
直接経費
間接経費
合計
平成 1 6 年度.
3 ,5 0 0 ,0 0 0
0
3 ,5 0 0 ,0 0 0
平成 1 7 年度
2 ,8 0 0 ,0 0 0
0
2 ,8 0 0 ,0 0 0
平成 1 8 年度
3 ,0 0 0 ,0 0 0
0
3 ,0 0 0 ,0 0 0
総計
9 ,3 0 0 ,0 0 0
0
9 ,3 0 0 ,0 0 0
以上
平成19年3月
研究代表者岩崎秀樹
(広島大学大学院教育学研究科教授)
Vll
International
Cooperation
Project
Towards the Endogenous
Development
of Mathematics
Education
Final
Report
Preface
The principle
of "Education for all (EFA)" has been a common goal in educational
development
among United Nations agencies and developed countries as well as developing
countries since
the World Conference on Education for All at Jomtien, Thailand
in 1990. In World Education
Forum at Dakar, Senegal
in 2000,
however, it was pointed
out that the various issues
surrounding
the education
in developing
countries
remain unsolved.
While the attempt of
evaluation
has been made by establishing
specific
goals for action in various organizations,
a
continuous
and foundational
research is essential
to build the framework of evaluation
as it
ought to be. While the significance
to focus on reality of "lessons" is often mentioned in
international
cooperation
in education,
it is doubtful
that we are fully aware of the reality of
lessons in developing
countries and its historical
and sociocultural
background.
A framework of evaluation
in international
cooperation
in education
is greatly affected by the
educational
experience
in each country. And also, evaluation
indicators
tend to be used for
symptomatic treatments. While "Lesson study", which put emphasis on the process of teaching
and learning, is regarded
as an important measure in international
cooperation
in education by
Japan, the framework of project evaluation
is still based on a behavioristic
way of ex ante and
ex post analysis.
We never understate
the significance
of clarity,
objectivity,
systemicity,
quantification,
and efficiency
advocated in the behavioristic
evaluation.
We, however, would
like to make aware of and place greater value on the essence in international
cooperation
in
education,
such as endogenousness
of educational
body, autonomy of teaching profession,
and
creativity
of children.
In developing
an alternative
framework of evaluation
from a
multidimensional
perspective,
it is important
for us to have a consciousness
of unmeasurable
factors and think of both the effectiveness
and the limitation
of certain methodology.
In order to
appropriately
evaluate
intricate
and multitiered
lessons,
it is urgent issue to develop
an
alternative
framework of evaluation
in which the behavioristic
evaluation
become part of the
indicators.
This research project aims at developing
the indicators
and procedure with the emphasis on the
process of teaching
and learning
for some participating
countries,
valuing "description"
and
"interpretation"
rather than behavioristic
"measurement" and "prediction".
When we address the
reality of children's
achievement and teaching-learning
environment in mathematics
education,
it is inevitable
for us to focus on historical
and sociocultural
aspects of education.
Since the
living educational
information
in developing
countries
with a great variety of historical
and
sociocultural
background
is essential
in this research project, we implemented
field surveys in
2004 and 2005. In addition,
we held the following
international
workshops to share and
accumulate the result of the field surveys to clarify and discuss wide-ranging
issues and the
solutions.
i) The first international
workshop towards the endogenous development of mathematics education,
Date and venue: March 15-17, 2005, in IDEC, Hiroshima University,
i ) The second international
workshop towards the endogenous development of mathematics education,
Date and venue: December 11-13, 2005, in IDEC, Hiroshima University.
i i) The international
symposium on international
cooperation towards the endogenous development of
mathematics education,
Date and venue: January 14, 2007, in the Institute for International
Cooperation, Tokyo.
Here, I would like to make a most cordial acknowledgment
to those who contributed
to this
three-year project. Fist of all, I would like to express my sincere gratitude
for the professional
assistance
by Professor Emeritus Alan J. Bishop, Monash University,
Australia.
There could not
have been progress without his insightful
messages.
Vlll
International
Cooperation
Project
Towards the Endogenous Development
of Mathematics
Education
Final
Report
The third international
meeting mentioned above was held to ensure the result of our three-year
activities
as the international
symposium in cooperation
with the Center for Research on
International
Cooperation
in Educational
Development (CRICED),
Tsukuba University,
under
the auspices of the Ministry
of Education,
Culture, Sports, Science and Technology;
Japan
International
Cooperation
Agency (JICA);
Japan Society of Mathematics
Education;
and Japan
Society for Science Education.
It is indeed an honor for us to have the keynote speech, "The
world role of culture in mathematics
education" by Professor Norma C. Presmeg, Illinois
State
University.
This final report is mainly based on the reports presented in the international
symposium
2007. 1 would like readers to go through with interest the full report of this research project.
in
Lastly, I greatly appreciate
efforts made by all the research members towards the progress of
this research
project,
with regards to implementing
of the field surveys, holding
of the
international
workshops and the symposium, and compiling
the final report.
Hideki IWASAKI
Professor
Graduate School of Education
Hiroshima University,
March 2007
IX
Chapter
Research Background
1
and Methodology
International
Cooperation
Project
of Mathematics
Education
Final
Chapter
1
Research
1.1
Research
1.1.1
Background
Background
Background
Report
and Methodology
and Objective
Since the World Conference on Education for All at Jomtien, Thailand
in 1990, international
organizations
and individual
countries have been seriously
engaged in improving the quality of
basic education
in developing
countries. In this global context, since 2004, our research group
has been engaged in identifying
the status of mathematics
education from the viewpoints
of
children's
learning and its several interrelated
factors as well as teaching in the participating
countries through collaborative
and joint research.
In this research project, we especially
attach importance to the context of teaching and learning
in mathematics
education. As Figure 1 shows below (UNESCO, 2005), teaching
and learning
process is in the center of whole education quality framework. And also, the involvement
of
several interrelated
factors and interpretations
make it difficult
to define and realize quality
education, which is a major task since 1990's conference. It is a very urgent and long-term issue
to grasp the status of teaching and learning and its context.
Figure 1. A framework for understanding
education
Enabling
inputs
Teaching
å
å
å
å
Leaner
characteristics
å
å
å
å
å
Aptitude
Perseverance
School readiness
Prior knowledge
Barriers
to learning
quality
and learning
Leaning time
Teaching methods
Assessment, feedback,
Class size
å Teaching and learning materials
I Physical
infrastructure
and facilities
I Human resources: teachers, principals,
inspectors,
supervisors,
and administrators
å School governance
I
:
å Economic and labour
market conditions
in
the community
å Sociocultural
and religious
facto rs
å [Aid strategies]
I Educational
knowledge
and support infrastructure
I Public resources available
for education
I Competitiveness
of
the teaching
profession
on the labour market
I National
governance and
management strategies
1.1.2
incentives
Context
I
I Literacy,
numeracy,
and life skills
å Creative and
emotional
skills
å Values
å Social benefits
I
I Philosophical
standpoints
of the teacher and the learner
I Peer effects
I Parental
support
I Time available
for
schooling
and homework
I National
standards
I Public expectations
I Labour market demands
Globalization
Objective
This research
group has conducted
an international
comparative
study with above-mentioned
International
Cooperation
Project
Development
of Mathematics
Education
Final
background
(1)
for the period
2004-2006.
The specific
objectives
of this project
Report
are as follows:
To compare and analyze the children's
achievement
in mathematics
education
in the
participating
countries with both an internationally
recognized
test such as TIMSS and an
alternative
test. Simple comparisons,
however, cannot be made to decide whether one is
better than any other country.
(2) To exchange views and information
on the issues found in (1). Through this process,
several factors which were unconsciously
embedded in teaching
and learning would be
sought for an alternative
framework of the evaluation of children's
achievement.
(3)
1.2
Finally,
to establish
a professional
various participating
countries
in
education especially
in mathematics
In this research project, we build the
development" (Tsurumi and Kawata,
Research
and long-term relationship
or network among the
the field of curriculum
development
and teacher
education through the process of knowledge sharing.
foundation to find the way towards "the endogenous
1989) in mathematics
education.
Members
The list of the research project members both in Japan and overseas is provided below. Apart
from the members on the list, there are three special advisors-renowned
professors in this
field-whose valuable contributions
added another dimension to this report.
Table
1. Members
N A M E
H id e k i Iw a sa k i
A tsu m i U e d a
T ak u y a B a b a
M a s am i Iso d a
H id e k i M a ru y a m a
H isa sh i K u w a y a m a
H iro y u k iN in o m iy a
K a z u m a O h a ra
S a to sh iN a k a m u ra
U d d in M D M o h s in
T o y o m i U c h id a
C h ik ara K in o n e
L e v i E lip a n e
in Japan
(As of March,
A fi liatio n
F a c u lty o f E d u c atio n ,
H iro sh im a U n iv e rs ity
ID E C ,
C R E IC E D ,
T s u k u b a U n iv e rsity
N a tio n a l In stitu te o f
E d u c a tio n a l P o lic y
R e se a rc h
H iro sh im a U n iv e rs it
F ac u lty o f E d u c a tio n ,
S a ita m a U n iv e rsity
N aru to U n iv e rsity o f
E d u c a tio n
ID E C ,
H iro sh im a U n iv e rsity
ID E C ,
H iro s h im a U n iv e rsity
ID E C ,
H iro sh im a U n iv e rsit
ID E C ,
S a ita m a U n iv e rs ity
2007)
P o sitio n
P ro fe sso r
P ro fe sso r
A sso c iate p r o fe sso r
A sso c iate p ro fe sso r
R o le
O v e ra ll m a n a g er,
C h in a
C h in a
C o o rd in ato r,
B a n g la d e s h
T h a ila n d
R e se a rch e r
D e v e lo p m e n t o f
re se a rc h to o ls, G h a n a
R e se a rch A s so c ia te
C o o rd in a to r, G h an a
A sso c iate p ro fe ss o r
P h ilip p in e s
A sso c ia te p ro fe ss o r
T h a ila n d
R e s ea rc h e r
Z a m b ia
G ra d u ate S tu d e n t
B a n g la d e sh
Z a m b ia
J ap a n
G ra d u ate S tu d en t
P h ilip p in e s
International
Cooperation
Project
of Mathematics
Education
Final
T a b le 2 . M e m b e rs O v e rs e a s (A s o f M a rc h , 2 0 0 7 )
N A M E
A fi lia tio n
P o s itio n
A .H .M . M o h iu d d in
N A PE
(T h e N atio n a l S p e c ia list
A c a d e m y fo r P rim a ry
E d u c a tio n )
D a i Q u in
In n e r M o n g o lia
P ro fe s s o r
U n iv e rs ity o f E d u c atio n
J in K a n g b ia o
ID E C ,
G rad u a te
H iro sh im a U n iv e rsity
E rn e st D a v is K o fi
U n v e rs ity o f C a p e C o a st L ec tu rer
G h a rt ey A m p ia h
U n v e rs ity o f C a p e C o a st S e n io r L e ctu re r
N arh F ra n c is
M o u n t M ary C o lle g e o f
L e c tu re r
E d u c a tio n
M ilag ro s D . Ib e
U n iv e rs ity o f
E m e ritu s P ro fe ss o r,
th e P h ilip p in e s-D ilim a n
D ean
M a itre e In p ra s ith a
K h o n K ae n U n iv e rs ity
A sso c ia te P ro fe sso r
B e n try N k h ata
U n iv e rsity o f Z a m b ia
L e ctu re r
Table 3. Special
N A M E
A la n J . B ish o p
W ee T io n g S e a h
N o rm a P re sm e g
1.3
Advisor
Report
R o le
B a n g la d e sh
C h in a
C h in a
G h ana
G ha na
G ha na
P h ilip p in e s
T h a ila n d
Z a m b ia
(As of March, 2007)
A ffilia tio n
M o n a sh U n iv e rsity
M o n a s h U n iv e rsity
Illin o is U n iv e rs ity
P o s itio n
P ro fe sso r E m e ritu s
L ec tu re r
P ro fe sso r
C o u n try
A u stra lia
A u stra lia
U SA
Contents of Report
This report consists of four sections, namely, Research background and Methodology, Special
Contribution,
Country Report, and Discussion and Future Issues. In Special Contribution,
research trends and issues have been intensively discussed.
With regard to the objectives, Country Report is the primary section of this report. It presents
the data in the format that conforms to our prior agreement, which is as follows:
(Composition
of Country Report )
•E Basic Information
>
School Curriculum
>
SchoolingAge
>
School Calendar and Examination
>
Promotion System
>
Medium of Instruction
>
Class Organization
>
Period of Survey
>
Target Schools
International
Cooperation
Project
Development
of Mathematics
Education
Final
Results
from the Field
Report
Survey
>
Coverage of the Achievement
>
Results
of the Achievement
>
Results
of the Pupils'
Test in Curriculum
Test
Questionnaire
and Interview
Discussion
The Country Report presents at first basic data such as schooling
age and promotion system,
which we think have given impact on pupils' learning. And result and discussion
are presented
afterwards. The result presents the data collected
through the field survey. And the discussion
part presents some points of interests.
The "Future Issues" constitutes
also very crucial part of this research.
It provides
the
comprehensive discussion
over the whole research and projects regarding some important issues
for future activities.
This cycle "projection
-research (field survey) - discussion"
gives us
foundation for our ultimate goal of the Endogenous Development of Mathematics Education.
Besides these,
summarized.
in order to strengthen
1.4
Research
1.4.1
TargetSchoolsand
the discussion,
some past researches
will
be reviewed
and
Methodology
Grade
Average schools from both urban and rural areas were chosen after consultation
with local
experts. Basically,
we conducted the field survey in the same schools (for 2004 and 2005). In
some countries,
where language problem appeared to influence
the result, an additional
comparative survey was conducted (e.g., no explanation vs. explanation in local language).
Grade four was chosen as the original focus in the planning of this research. The reasons were
as follows: 1) TIMSS tools were available for grade four in the first year survey. 2) Grade four
was perceived as a transitional
stage from the concrete operational
period to the formal
operational
period. 3) Besides
this, countries
like Bangladesh
had five years for primary
education and the final year was influenced by other factors such as the scholarship
examination.
However, there were some exceptions in target grade due to the timing of each field survey in
participating
counties.
1.4.2
ResearchTools
in 2004
The research tools such as a questionnaire
and tests were prepared
revised through discussion
among co-researchers.
by the Japanese
team and
As for Pupils'
achievement test, the prototype was designed on the basis of items in TIMSS
1995 and 2003. Table 4 shows the structure of test items by content domains in the pupils'
achievement test. The ratio in numbers of the items by content domains in our tests was nearly
the same as in TIMSS, such as 40% for Number, 15% fof Patterns, Relations
and Functions,
21% for Measurement, 14% for Geometry, and 10% for Data. Some items were, however,
excluded from TIMSS items, as this survey did not focus on complicated
problem solving, but
basic learning achievement. As a result, the ratio in numbers of the items by cognitive
domains
in TIMSS 2003 was not preserved in this project. The ratio of multiple choice question to free
answer question were not maintained either, such as 67% for multiple choice question and 33%
for free answer question, while 57% for the former and 43% for the latter in TIMSS 2003.
International
Cooperation
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Towards the Endogenous Development ofMathematics
Education
Final
Table 4. No. of Items in Mathematics
N u m b er
P a tte rn s ,
R e latio n s ,
an d
F u n c tio n s
M e a su re m e n t
G e o m etry
D a ta
T o ta l
Test Items by Content
Ite m s fro m T IM S S 2 0 0 3
(In v o lv in g 3 7 'T ren d ' ite m s (5 1% o f fu ll ite m s ))
K n o w in g
S o lv in g
U s in g
F a c ts an d
R o u tin e
R e a so n in g
C o n c e p ts
P ro ce d u re
P ro b le m s
8
8
6
1
Report
Domains
Ite m s fro m
T IM S S 19 9 5
T o ta l
ex cept
'T re n d ' Ite m s
6
2 9 (4 0 % )
0
2
2
1
6
l l( 15 % )
5
4
1
0
3
4
2
1 5 (2 1% )
0
1 7 (2 4 % )
0
l l ( 15 % )
2
1
1
6 (8 % )
3
3
1 7 (2 4 % )
3
2 1(2 9 % )
1 0 ( 14 % )
7(10% )
7 2 ( 10 0 % )
Table 5 shows the main topics in the mathematics achievement test items. Especially,
some free
response questions
were set for Fraction and Decimals to investigate
how pupils understand
concepts. Mathematical
content of each item were basically
preserved to examine the process
and result of TIMSS critically
from the perspective
of endogenous development of mathematics,
while the setting of some questions were modified to exclude culturally
unfamiliar
terms in
some countries (not all the participating
countries).
Table 5 Contents
of Mathematics
C o n ten t D o m a in s
N u m ber
P a tte rn s, R e latio n s , a n d F u n c tio n s
M e a su re m e n t
G e o m etry
D a ta
T o ta l
Test Items
M a in T o p ic
W h o le N u m b e rs
F ra c tio n s a n d D e c im a ls
R atio , P ro p o rtio n a n d P e rc e n t
P a tte rn s
E q u a tio n s a n d F o rm u la s
R e la tio n sh ip s
A ttrib u te s an d U n its
T o o ls , T e ch n iq u e s a n d F o rm u la s
L in e s a n d A n g le s
T w o - a n d T h re e -d im e n sio n a l S h ap e s
C o n g ru e n c e an d S im ila rity
L o c a tio n s a n d S p atia l R e la tio n s h ip s
S y m m etry a n d T ran sfo rm atio n s
D a ta R e p re se n ta tio n
D a ta In te rp re ta tio n
N o . o f Item s
18
10
1
4
4
3
8
7
1
2
1
2
4
4
3
72
Questionnaire for pupils were also based on TIMSS research tools. It was slightly modified to
suit to the situation
in developing
countries. It focused on affective aspect of pupils'
achievement and their learning environment including out-of-school surroundings.
Both mathematics achievement tests and questionnaires are attached to this report.
International
Cooperation
Project
Development
of Mathematics
Education
Final
1.4.3
ResearchTools
in 2005
In 2005, the research tools such as test and interview items
understanding
about fraction. The reason was that fraction
in abstract
mathematics,
so that it would be suitable
for
transitional
stage from the concrete operational
period to the
In principle,
prohibited
to
English
test,
conditions
of
Report
were designed to evaluate children's
had been perceived as the first topic
grade four pupils who were on a
formal operational
period.
any additional
explanation
which affected how to get the correct answer was
give to pupils. However, since we implemented mathematics achievement test, not
the following
three ways of implementation
were allowed depending
upon the
participating
countries.
(a)
Neither
of reading in mother tongue nor additional
answer (Standard method)
(b)
Reading of test item in mother tongue, but no additional
the answer (Alternative
method I)
(c)
Both Reading of test item in mother tongue
express the answer. (Alternative
method II)
In case of(b) or (c), any additional
case (a), if possible.
sample
had better
explanation
of how to express
explanation
and additional
the
of how to express
explanation
be chosen for comparative
of how to
survey with the
As for the procedure for interview, we selected ten pupils, top five pupils and bottom five pupils,
according to the result of the latest end-of-term examination. We used the Newman procedure to
investigate
their understanding
about the questions and the strategy for problem-solving.
Both mathematics
achievement
tests and interview
items are attached to this report.
[Newman Procedure]
The Newman Procedure for analyzing errors on written mathematical
tasks. Since 1977, when
the Australian
educator M. A. Newmanpublished
data based on a system she had developed
for
analyzing
errors made on written tasks, a steady stream of research papers has been published
in
many countries in which data from many countries have been reported and analyzed along line
suggested by Newman.
Analyses of data based on NewmanProcedure have drawn special attention to (a) the influence
of language factors on mathematics
learning
(b) the inappropriateness
of many 'remedial'
mathematics programs in schools in which there is an over-emphasis on the revision of standard
algorithms.
According to Newman Procedure with a one-step written problem has to read the problem, then
comprehend what he/she read, then carry out the transformation
from the words to the selection
of an appropriate
mathematical
'model', then apply the necessary process skills, then encode the
answer.(see Figure 2)
Errors due to the form of the question would appear to be essentially
different
from errors in the
other categories
shown in the above because the source of difficulty
resides fundamentally
in
the question itself rather than in the interaction
between the question and the person attempting
it. This difference
is indicated
by the category labeled 'Question
Form' being placed beside the
five-step hierarchy.
Two other categories
'Carelessness'
and 'Motivation'
have also been shown
as separate from the hierarchy
although,
as indicated,
these types of errors can occur at any
stage of problem-solving
process. A careless error, for example, could be a reading error, or a
comprehension
error, and so on; similarly,
someone might decide not to try to get the correct
answer for a problem even after he has read and comprehended it satisfactorily,
and carried out
an appropriate
'transformation'
International
Cooperation
Project
Towards the Endogenous Development of Mathematics
Education
Final
Figure
Report
2. The Newman Procedure
C h aracteristics of
th e q u estion
Interaction b etw een th e qu estion
an d th e person attem p tin g it
R ead ing
C om p reh ension v * ,
c arelessness
T ransform ation
M otiv ation 1
PE rocess
ncodingS kills T he N ew m an hierarchy for one-step
Q uestion F orm
verbal m athem atical problem s
Newmanrecommendedthat the following 'questions' or requests be used in interviews
carried out in order to classify students' errors on written mathematical tasks:
1.5
that are
1.
Please read the question to me. If you don't know a word leaves it out. (Reading)
2.
Tell me what the question is asking you to do. (Comprehension)
3.
Tell me how you are going to find the answer. (Transformation)
4.
Show me what to do to get the answer. Tell me what you are doing as you work.
(Process skills)
5.
Nowwrite down the answer to the question.
Plan of Activity
FIRST YEAR: To conduct a preliminary study in order to generally
mathematics education and its problems in each participating
country.
describe
the status of
(1 ) To contact the local expert
The following countries were selected: Ghana and Zambia in Africa; Bangladesh in South Asia;
Myanmar,T hailand, and Philippines
in Southeast Asia; and China in East Asia. In each country,
w eselected a curriculum developer or a teacher trainer. Apart from this, we presupposed that
everybody could be contacted by email.
International
Cooperation
Project
of Mathematics
Education
Final
(2) To discuss
the research
Report
methodology
Weprovided the first draft plan for methodology
and then, for further refinement, discussed
it
with the local experts. Although,
on the whole, we desired to describe the general features of
mathematics
education
in each country, a comparative evaluation
of the same was not our
intention.
We analyzed the points of similarity
and difference.
Further, in order to describe the
average status of each country, we compared contrasting entities,
e.g., urban vs. rural.
(3) To conduct
the preliminary
study
Werequested each local expert to conduct the preliminary
study. They were allocated
a budget,
which enabled them to conduct the preliminary
study, along with video shooting,
translation
of
the questionnaires,
execution of the survey, and its analysis.
(4) The Japanese experts
confirm the data collected.
visited
different
countries
(5) International
workshop
An international
for the following
workshop
year.
(6) Compilation
of the report of the first year study
The results
Figure
was held to discuss
of the first year preliminary
for quality
control
and share information
study were compiled
and went on site to
and problems,
and to plan
as a report.
3. Plan of Activity
2 00 4
9
10
2 00 6
2 00 5
ll
12
1
2
3
4
5
6
7
8
9
10
ll
I 蝣蝣 I 蝣蝣
蝣IB !蝣mI^^H蝣H
Pr
el
i
mi
nar
y
I
St
4) 蝣 I 蝣蝣I 蝣I m! 蝣蝣!* 蝣! ! 蝣 !蝣
W udy3)
orkshop&5)
Report 6)
12
*
1
2
3
蝣蝣
蝣
蝣 I *
SECOND YEAR: To conduct the main study and to analyze and discuss the findings.
(1) To discuss the research methodology
Based upon the discussion
during the first international
workshop, the Japanese experts
developed a draft of research tools and discussed it with the local experts with regards to the
main study.
(2) To conduct the main study
The local and Japanese experts conducted the main study. The Japanese data was also collected
for the discussion of the first year.
(3) International
workshop
The workshop discussed and shared information and the results of the research.
(4) Compilation
of the report of the second year study
International
Cooperation
Project
of Mathematics
Education
Final Report
THIRD YEAR: To write a summary report of the three-year
three years.
(1) To conduct a complementary
If necessary,
a complementary
of the final
and to plan for the next
study
study will be done.
(2) To conduct an international
(3) Compilation
activity
seminar in Tokyo, Japan
report
Referen ces
Newman, M. A. (1977),
An Analysis
Tasks,
for Educational
Victorian
Institute
of Sixth-Grade
Research Bulletin,
Tsurumi, K. & Kawata, T. edits
Japanese).
(1989)
UNESCO (2005)
Report, Paris.
EFA Monitroing
Pupils'
Endogenous
Error on Written
Mathematical
39, 3 1 -43.
development,
Tokyo University
Press
(in
Chapter
2
Country Report
International
Cooperation
Project Towards the Endogenous Development ofMathematics
Education
^ FinalReport
Chapter
2 Country
Report
2.1 Bangladesh
2.1.1
School
UDDIN MD.MOHSIN
TAKUYA BABA
Open University,
Hiroshima
Bangladesh
University
Curriculum
Since 1992, a curriculum known as the Essential
Learning Continuum (ELC) comprising
53
competencies has been introduced at the primary education level in Bangladesh.
The ELC is a
listing
of competencies that serves as a guide for determining
what to teach and assess among
students at the primary level. Competency has been defined as "the acquired knowledge, ability,
and viewpoint when these could be applied in real life at the right time" (translated
from the
original
mathematics
curriculum by the author).
These competencies
are based on the
psychological
maturity, age, and physical capability
of the children,
relate to the changes taking
place in our society, demands for the future, existing physical
facilities
in schools, and the
preparation of teachers at the school level. Children are expected to acquire these competencies
in the five-year term of primary education;
these competencies are referred to as the "terminal
competencies of primary education." It is claimed that this goal provides the all-round profile or
vision of development for a child who completes the five-year cycle of primary education.
From among the 53 competencies,
5 are mathematical.
These are as follows:
To gain the basic ideas of number and to be able to use them.
To know the 4 fundamental
To apply the simple
everyday life.
operations
methods
and to be able to use them.
of computation/calculation
To know and use the units of money, length,
To know and understand
2.1.2
(1)
Basic
Schooling
geometrical
weight,
to problem-solving
in
measurement, and time.
shapes and figures.
Information
Age
The official age for beginning school is 6 years; however this varies from case to case. There is
no birth registration
system in Bangladesh.
Certain parents, particularly
in rural areas, are
unaware with regard to the age of their children. Occasionally,
they measure age according to
the height of the children.
(2)
School
Calendar
and Examination
The school academic year begins from January and ends in December. In general,
school terms in Government Primary School (GPS): The first term (January-April);
term (May-August),
and the third term (September-December).
Table 1 shows
calendar in 2005; shaded portions represent long holidays.
These holidays depend
ll
there are 3
the second
the school
on the lunar
International
Cooperation
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Education
Final Report
calendar
and vary with each year.
T a b le 1 . S c h o o l c a le n d a r in 2 0 0 5
J an .
F eb .
M a r.
A p ril
M ay
Ju ly
Jun e
A ug .
S ep .
O ct.
N o v.
D ec.
^^^^^^^H 蝣
l j^ S ^ S
T h e r e is n o n a t i o n a l l y e s t a b l i s h e d e x a m i n a t i o ns y s t e mf r o mg r a d e s 1 t o 4 a t t h e p r i m a r y l e v e l i n
B a n g l a d e s h . T h e r e is a t e r m e x a m i n a t i o na t t h e e n d o f e a c h t e r m i n individual schools.
M o r e o v e trh, e r e is a s p e c i a l n a t i o n a l e x a m i n a t i o ni n g r a d e 5 k n o w na s t h e p r i m a r y s c h o l a r s h i p
e x a m i n a t i o n E. v e r y G P S i s r e q u i r e d t o s e n d s t u d e n t s i n t h e t o p 2 0 p e r c e n t , w h oa r e c o m p l e t i n g
g r a d e 5 , f o r this e x a m i n a t i o n .M e r e l y 31.7 p e r c e n t o f t h e e x a m i n e e sp a s s e d t h e s c h o l a r s h i p
e x a m i n a t i o ni n 2001 ( D i r e c t o r a t e o f P r i m a r y E d u c a t i o n , D P E , 2 0 0 2 ) . T a b l e 2 s h o w s t h e s c h o o l
c a l e n d a r i n 2005; s h a d e d p o r t i o n s r e p r e s e n t t e r me x a m i n a t i o n s .
Table 2. School examinations
J an .
F eb .
M a r.
A p r il
M ay
Ju n e
Ju ly
A ug.
S ep .
O c t.
N ov.
D ec.
(3) Promotion
System
The promotion system in grades I and II is relatively
relaxed. To a certain extent, it is an
automatic promotion system. In grades III, IV, and V, it is compulsory for each student to give
an annual exam in their individual
schools and pass the exam. The pass percentage for each
subject is 33.
(4) Medium of Instruction
The medium of instruction
at the primary school level is Bangla. Bangla is the mother tongue
for a majority of the people (e.g., Muslim and Hindu) and the national language (official
language) in Bangladesh.
(5) Class Organization
At the primary level, teachers teach all subjects
teacher. Each lesson has a duration of 35 minutes.
in different
grades;
there is no subject-specific
A majority of the primary schools function with a two-shift schooling
system. The first shift is
from 9:30 am till 12:00 noon and is for grades I and II; the second shift is from 12:30 pm till
4:15 pm and is for grades III, IV, and V.
Schools that function withjust
one shift begin at 9:30
and II and till 4:15 pm for grades III, IV, and V.
am and continue till
1 : 15 pm for grades I
In two-shift schools, the weekly class load for a teacher is 46.6 lessons in rural schools and 43.5
lessons in urban schools. In one-shift schools, the weekly class load of a teacher is 39.5 lessons
in rural schools and 38.5 lessons in urban schools (The National
Academy for Primary
Education, NAPE, 2002).
12
International
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Table
3. School
hours
F ir st sh ift
H our
8
9
G ra d e
ll
12
13
G ra d e I a n d II
2.1.3 Results
(1)
10
S e c o n d sh ift
14
15
16
17
G ra d e III, IV an d V
from the First Year Field Survey
Survey Schedule
Table 4. Schedule
D ate
2 5 "1 N o v e m b e r, 2 0 0 4
2 6 tn N o v e m b e r, 2 0 0 4
2 7 m & 2 8m N o v e m b e r, 2 0 0 4
2 9 m & 3 0 m N o v e m b e r, 2 0 0 4
1 st & 2 na D e c e m b e r, 2 0 0 4
3 rd D e c e m b e r, 2 0 0 4
4 m D e ce m b er, 2 0 0 4
5 m D e ce m b er, 2 0 0 4
(2) Target
Schools
of data collection
A c tiv ity
A rriv a l in D h ak a , B a n g la d e sh
M y m e n sin g (N A P E )
N A P E (M e etin g fo r th e re se a rc h , q u e stio n n a ire
re v isio n , c o rre ctio n e tc .)
D a ta c o lle ctio n in U rb a n sc h o o l, M y m e n s in g
D a ta c o lle c tio n in R u ra l sch o o l, M y m en sin g
R e tu rn to D h ak a
S tay in D h a k a
D e p a rtu re fro m D h a k a , B a n g lad e sh
and Samples
In Bangladesh, governmentprimary
categorization is based on certain
Parent-Teachers Association (PTA),
playground, plantation, scout group,
schools are categorized into 4 levels-A, B, C, and D. This
factors such as School ManagementCommittee (SMC),
results of the students, results of scholarship examination,
etc. A majority of the schools belong to category B.
Fromamong1,249 GPS, 1 average GPS (B category school) was selected from an urban area
and 1 average GPS (B category school) was selected froma rural area in Mymensingh district
(Primary Education Statistics in Bangladesh, 2001). It must be noted that Mymensingh is one of
the 64 districts in Bangladesh.
Table 5. Location of schools
U rb an P rim a ry S c h o o l
R u ra l P rim a ry S c h o o l
S c h o o l lo c a tio n
C e n ter o f M y m e n s in g h c ity
A b o u t 3 0 k ilo m ete rs aw a y fr o m th e m a in c ity
The selected urban school is situated in the center of Mymensingh city and the rural one is
situated approximately 30 kilometers away from the main city. The former has a total of 6
teachers and the latter 5. These teachers have to teach all the subjects in different grades; there is
no subject-specific teacher at the primary level.
There are a total of 6 rooms in both schools; 5 rooms are used as classrooms and the remaining
as office-cum teachers' commonroom.The rooms used as classrooms are well decorated with
colorful pictures and figures from textbooks of social studies, science, mathematics, and
13
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Education
Final
geometry. Verses from the holy Quran and a selection
inside the classrooms.
Table
of morals are also written on the walls
6. Distribution
of sample
G rad e
B oy s
4
14
4
15
蝣^^ ^^^^ ^- ^ ^^ｻ l 2 9
U r b a n P r im a ry S c h o o l
R u r a l P r im a ry S c h o o l
T o ta l
Report
G ir ls
T o ta l
46
29
75
32
14
46
T a b le 7 . D is t r ib u t io n o f s t u d e n t s ' a g e
A ge
(Y e ars )
U rb a n
8
9
10
4
13
17
3
4
1
ll
12
13
14
A v e ra g e
N o . o f P u p ils
R u ra l
ll
3
8
T o ta l
15
16
25
4
8
1
4
1
1
9 .7
4
1 0 .2
2
5
10
(3) Results of Student Questionnaire
(a)About
how many books are there in your home?
Table
8. Distribution
of students'
books
(%)
10 0 -
N o
C ro s se d
0 -1 0
b oo ks
l l -2 5
b oo ks
2 6 -1 0 0
b oo k s
bo ok s
C o m b in e d
7 2 .0
1 4 .7
1 .3
5 .3
re s p o n s e
2 .7
o ut
4 .0
U rb a n sc h o o l
6 7 .4
2 1 .7
2 .2
4 .3
2 .2
2 .2
0 .0
6 .9
3 .4
6 .9
R u ra l sc h o o l
7 9 .3
3 .4
(b) Do you have any of these items at your home?
T a b le 9 . D is t r ib u t io n o f s t u d e n t s ' h o m e it e m s (Y e s , % )
C a lc u la t o r
T V
R a d io
D esk
Q u ie t
D ic t io n a r y
B o ok s
C o m p u te r
( E x c lu d in g
p la c e
te x t b o o k )
C o m b in e d
U rb a n sch o o l
R u ra l sch o o l
3 0 .7
6 0 .0
4 4 .0
6 0 .0
4 0 .0
1 8 ,7
4 9 .3
5 .3
3 4 .8
7 1 .7
4 3 .5
6 0 .9
4 1 .3
2 6 .1
5 2 .2
2 .2
2 4 .1
4 1 .4
4 4 .8
5 8 .6
3 7 .9
6 .8
4 4 .8
1 0 .3
14
International
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(c) How much do you agree with these statements
Legend: 1:agree a lot, 2: agree a little, 3: disagree
Table
1 0. students'
attitude
about learning mathematics?
a little, 4: disagree
a lot
towards
mathematics
A v e ra g e (% )
P u p ils ' P e r c e p t io n s
C o m b in e d
U rb a n
R u ra l
1 . 1 u s u a lly d o w e l l in m a th e m a tic s
1 .3
1 .1
1 .7
2 . 1 w o u ld lik e t o d o m o re m a t h e m a t ic s in s c h o o l
1 .5
1 .3
1 .7
2 .2
2 .5
1 .6
3 . M a th e m a t ic s is h a rd e r f o r m e th a n fo r m a n y o f
m y c la s s m a te s
4 . 1 e n j o y le a rn in g m a th e m a t ic s
1 .5
1 .4
5 . 1 a m j u st n o t g o o d a t m a th e m a t ic s
2 .2
2 .3
1 .5
2
6 . 1 le a r n th in g s q u ic k ly in m a th e m a t ic s
7 . 1 th in k le a rn in g m a t h e m a tic s w il l h e lp m e in m y
1 .6
1 .4
1 .8
1 .4
1 .3
1 .7
1 .5
1 .4
1 .5
1 .3
1 .2
1 .5
1 .4
1 .3
1 .4
d a ily lif e
8. I
need
m a th e m a t ic s t o
le a rn
o th e r
sch o o l
s u b j e c ts
9 . I n e e d to d o w e ll in m a t h e m a t ic s t o g e t in t o t h e
u n iv e r s ity o f m y c h o ic e
1 0 . 1 n e e d t o d o w e ll in m a t h e m a t ic s to g e t th e j o b
I w ant
(d) How often do you do these things in your mathematics
lessons?
Legend: 1: every or almost every lesson, 2: about half of the lessons,
4:never
Table
ll. students'
activities
in their
mathematics
3: some lessons,
lessons
A v era g e (% )
C o m b in e d
U rb a n
R u ra l
S t u d e n ts ' a c tiv it ie s
1 . I p ra c tic e a d d in g , s u b tr a c t in g , m u ltip ly in g , a n d d iv id in g
w it h o u t u s in g a c a lc u la to r
1 .6
1 .4
1 .7
2 . 1 w o r k o n f r a c t io n s a n d d e c im a ls
1 .4
1 .3
1 .6
3 . 1 m e a s u r e t h in g s in th e c la s s r o o m a n d a ro u n d th e s c h o o l
2 .9
2 .8
3 .1
4 . 1 m a k e t a b le s , c h a r ts , o r g r a p h s
2 .0
2 .2
1 .6
1 .7
1 .6
1 .7
1 .8
1 .8
1 .7
7 . 1 w o r k w ith o t h e r s tu d e n ts in s m a l l g ro u p s
1 .7
1 .8
1 .4
8 . 1 e x p la in m y a n s w e r s
1 .9
1 .8
2
9 . 1 lis t e n t o th e te a c h e r ta lk
1 .2
1 .1
1 .2
1 0 . 1 w o rk p r o b le m s o n m y o w n
1 .9
1 .7
2 .1
1 1 . 1 r e v ie w m y h o m e w o r k
1 .6
1 .5
1 .6
1 2 . 1 h a v e a q u iz o r t e s t
1 .7
1 .6
1 .9
1 3 . 1 u s e a c a lc u la to r
2 .9
2 .8
3 .2
5 . I le a r n
a b o u t sh a p e s su ch
a s c ir c le s , tr ia n g le s , a n d
re c ta n g le s
6 . I re la te w h a t I a m
le a r n in g in m a th e m a tic s to m y d a ily
liv e s
15
International
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Final Report
(e) How much do you agree with these statements
Legend: 1:agree a lot, 2: agree a little, 3: disagree
Table 12. students'
attitude
about your school?
a little, 4: disagree
a lot
towards
school
and teacher
A v e r a g e (% )
P u p ils ' p e r c e p t io n s
C o m b in e d
U rb an
R ural
1. 1 lik e b e in g in s c h o o l
2 . 1 th in k th a t st u d e n t s in m y s c h o o l tr y t o d o th e ir b e s t
1 .1
1 .1
1 .2
1 .4
1 .3
1 .4
3 . 1 th in k th a t t e a c h e r s in m y s c h o o l c a r e a b o u t th e st u d e n t s
1 .4
1 .5
1 .5
1 .3
1 .2
1 .4
4 .1 th in k th a t t e a c h e r s in m y s c h o o l w a n t st u d e n ts t o d o
th e ir b e s t
(f) On a normal school day, how much time do you spend before or after school
each of these things?
Table 13. students'
activities
before or after school (%)
N o t im e
L e s s th a n
1-2 h o u rs
1 ho ur
1 . 1 w atc h T V
2 .
I
p la y
or
t a lk
4 6 .7
2 6 .7
M
o r e th a n 2 b u t
doing
4 o r m o re
le s s th a n 4 h o u r s
h o u rs
1 .3
5 .3
9 .3
w ith
3 2 .0
4 6 .7
9 .3
1 .3
4 .0
3 .I d o job s at ho m e
2 9 .3
2 8 .0
1 3 .3
1 0 .7
9 .3
4 . 1 p la y s p o r ts
3 4 .7
3 6 .0
1 7 .3
1 .3
1 .3
f r ie n d s
5 .
I
re a d
a
b ook
fo r
4 1 .3
2 1 .3
1 7 .3
5 .3
6 .7
6 . 1 u se c o m p u te r
3 4 .7
8 .0
5 .3
1 4 .7
1 6 .0
7 . 1 d o h o m ew o rk
2 4 .0
3 0 .7
1 3 .3
5 .3
1 2 .0
e njo y m en t
(g) Including
yourself,
how many people
Table 14. Distribution
F a m ily m e m b e r
live in your home?
of students'
family
member
P u p ils ' D is tr ib u tio n ( % )
C o m b in e d
U rb a n
R u ra l
2 -3 p e rso n s
6 .7
1 .7
1 .7
4 -6 p e rso n s
5 0 .7
1 .9
1 .8
7 -9 p e r s o n s
9 .3
2 .5
2 .1
1 0 - 1 2 p ers o n s
6 .7
2
1 .6
2 0 .0
2 .2
1 .7
M
o r e th a n 1 2 p e r s o n s
(4) Results of Achievement Test
(a) The distribution
of right answers
iM ^ ^ H ^ ^ ^ ^ ^ ^ fi j A v era g e
of students'
right
answer (%)
B o ys
G irls
H ig h e st
L o w e st
C o m b in e d
3 3 .9
5 2 .1
1 5 .1
3 6 .8
3 5 .1
3 8 .6
3 3 .1
U rb an sc h o o l
3 6 .0
5 2 .1
2 1 .9
R u ra l sc h o o l
2 9 .1
3 1 .7
2 6 .4
4 2 .5
1 5 .1
16
International
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Education
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of students'
A g e (y ea rs )
(b) Analysis
right
answer according
C o m b in e d
U rb an sc h o o l
R u ra l s ch o o l
8
9
2 1 .8
2 4 .8
2 0 .7
2 8 .1
3 0 .5
1 8 .0
10
2 5 .1
2 3 .7
2 1 .5
ll
2 3 .9
1 7 .7
12
2 8 .4
3 0 .0
2 6 .8
13
2 5 .0
3 0 .0
2 0 .0
14
2 2 .4
2 3 .5
1 8 .0
1 8 .0
of the results with regard to content domains
Table 17. Average rates of correct answer by content
M u ltip ie c h o ic e
N o .o f
R ate o f
S h o rt a n sw er
N o.o f
R a te o f
q u e stio n s
c o rre c t
C o n ten t d o m a in
q u e stio n s
c o rre ct
N u m b er
M e a su re m e n t
A lg eb ra
D a ta
P atte rn s , re la tio n s ,
19
12
4
4
5
an sw e rs (% )
4 2 .9
4 0 .4
2 9 .7
5 5 .0
3 3 .3
10
3
1
4
1
an d fu n c tio n s
G e o m etry
T o ta l
4
48
2 2 .3
3 7 .2
25
Table
to age
A v e r a g e r ig h t a n s w e r ( % )
N o .of
Q u e stio n s
a n sw e rs (% )
3 7 .4
3 6 .4
2 3 .7
4 3 .1
2 7 .7
1 9 .3
1 6 .3
10
73
2 0 .5
3 1 .4
U rb a n s ch o o l
S h o rt
c h o ic e
4 5 .4
M e a su r e m e n t
c o rre ct
29
15
5
8
6
6
M u lt ip le
T o ta l
R ate o f
a n sw e rs (% )
2 6 .9
2 0 .4
0 .0
3 1 .3
0 .0
1 8. Average rates of correct answer of content
by urban and rural school (%)
C o n t e n t d o m a in s
domain
domain
R u ra l sc h o o l
T o ta l
M u lt ip le
S h o rt
T o ta l
a n sw e r
3 1 .3
4 0 .5
c h o ic e
3 9 .1
a n sw e r
2 0 .0
3 2 .4
4 1 .7
1 7 .4
3 6 .8
3 9 .1
2 5 .3
3 6 .3
A lg e b r a
3 0 .4
0 .0
2 4 .3
2 8 .4
0 .0
2 2 .7
D ata
5 7 .1
4 2 .9
50
5 1 .7
1 2 .9
3 2 .3
P a t te rn s , re la t io n s , a n d
4 1 .7
0 .0
3 4 .7
2 0 .0
0 .0
1 6 .6
2 4 .5
2 1 .7
2 2 .8
1 8 .9
1 5 .5
1 6 .9
N um b er
F u n c t io n s
G e o m e tr y
17
International
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Project
Education
Final
2.1.4 Results
(1) Survey
from the Second
Year Field
Report
Survey
Schedule
Table
19. Schedule
A ctiv ity
A rriv a l in D h ak a , B an g lad e sh
D a te
9 th N o v e m b e r, 2 0 0 5
1 0 th N o v e m b e r, 2 0 0 5
1 1th & 12 th N o v e m b e r, 2 0 0 5
13 th & 14 th N o v e m b er, 2 0 0 5
1 5th & 16 th N o v e m b er, 2 0 0 5
1 7 th N o v e m b e r, 2 0 0 5
(2) Target
Schools
of data collection
and Samples
Table
M y m e n sin g (N A P E )
N A P E (M e e tin g fo r th e re se arc h , q u e stio n n a ire
rev is io n , c o rre c tio n etc .)
D a ta co lle ctio n in U rb a n sc h o o l, M y m e n sinj
D a ta co lle ctio n in R u ra l sc h o o l, M y m e n sin g
D e p artu re fr o m D h a k a , B a n g la d e sh .
20. Location
of schools
S ch o o l lo ca tio n
C e n ter o f M y m e n sin g h c ity
A b o u t 3 0 k ilo m e te rs aw a y fro m th e m a in c ity
U rb a n P rim a ry S c h o o l
The selected urban school is situated in the center of Mymensingh city and the rural one is
situated approximately 30 kilometers away from the main city. There are a total of 6 teachers in
the former and 5 teachers in the latter. These teachers have to teach all subjects in different
grades; there is no subject-specific teacher in these schools.
There are a total of 6 rooms in both schools, where 5 rooms are used as classrooms and the
remaining as office-cum teachers' commonroom.The rooms used as classrooms are well
decorated with colorful pictures and figures from textbooks of social studies, science,
mathematics, and geography. Verses from the holy Quran and a selection of morals are also
written on the walls inside the classrooms.
[^
^
^
蝣
^
蝣蝣
Table
21. Distribution
蝣
i
G ra d e
5
5
U r b a n P r im a r y S c h o o l
R u r a l P r im a r y S c h o o l
T o ta l
(3) Results
of Achievement
(a) Overview of the result
Table
U rb a n S c h o o l
R u r a l S ch o o l
A ll
B oy s
8
ll
19
of sample
G ir ls
16
12
28
A ge
10 - 13
10 -14
10 -1 4
To
2
2
4
Test
22. Results
of the achievement
test (%)
A v e ra g e
5 1 .7 %
3 3 .7 %
B oy s
5 5 .3 %
3 3 .8 %
G ir ls
4 9 .9 %
3 3 .6 %
H ig h e st
68 %
53 %
L o w e st
3 4%
4 2 .9 %
4 2 .8 %
4 2 .9 %
6 8%
1 1%
18
1 1%
ta l
4
3
7
International
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Project
Education
Final
(b) Question-wise
achievement
of pupils
Table 23. Question-wise
in percentage
achievements
of pupils
S c h o o l in u r b a n a r e a
C o m b in e d
in percentage
S c h o o l in r u r a l a r e a
C o m b in e d
B oy s
G ir ls
Q l (D
9 1 .6 %
100 %
8 7 .5 %
3 0 .4 %
5 4 .5 %
8 .3 %
(2 )
8 9 .5 %
9 3 .7 %
8 7 .5 %
6 0 .8 %
6 3 .6 %
5 8 .3 %
Q 2
5 4 .1 %
3 7 .5 %
6 2 .5 %
0%
0 %
0%
Q 3 ( l)
9 6 .5 %
100 %
9 4 .7 %
4 0 .5 %
3 6 .3 %
4 4 .4 %
Q 3 (2 )
7 9 .1 %
7 5%
8 1 .2 %
2 6 .8 %
6 .1 %
4 5 .8 %
Q 4 ( l)
9 2 .7 %
10 0 %
89%
6 9 .5 %
4 3 .1 %
9 3 .7 %
Q 4 (2 )
0%
0 %
0%
1 8 .4 %
2 9 .5 %
8 .3 %
Q 4 (3 )
3 3 .3 %
3 7 .5 %
3 1 .2 %
1 6 .3 %
2 5%
8 .3 %
Q 5 (l )
3 5 .4 %
50%
2 8 .1%
13 %
9 .1 %
1 6 .6 %
Q 5 (2 )
2 .1 %
0%
3 .1 %
2 1 .7 %
4 5 .4 %
0%
Q 5 (3 )
6 .2 %
6 .2 %
6 .2 %
1 9 .5 %
9 .1%
2 9 .1%
Q 6 (l)
9 4 .7 %
9 6 .8 %
9 3 .7 %
3 8%
3 7 .5 %
3 8 .5 %
Q 6 (2 )
5 1 .5 %
9 8 .4 %
2 8 .1 %
2 8 .2 %
1 9 .3 %
3 6 .4 %
Q 6 (3 )
0 %
0%
0 %
13%
9 .1%
1 6 .6 %
0 7
2 7 .1 %
1 8 .7 %
3 1 .2 %
1 5 .2 %
3 1 .8 %
0%
0 8
5 8 .3 %
5 0%
6 2 .5 %
6 9 .5 %
8 1 .8 %
5 8 .3 %
Q 9
6 0 .4 %
6 2 .5 %
5 9 .3 %
5 8 .6 %
5 4 .5 %
6 2 .5 %
Q lO
1 2 .5 %
0%
1 8 .7 %
8 .6 %
1 8 .1 %
0 %
(4) Results
of Interview
using
B o ys
Report
G ir ls
Newman Procedure
Weselected 1 0 students for an interview on the basis of their mathematics exam result. Five of
them acquired a good grade in mathematics and the other 5 did not. The results of the students'
interview are presented in the following tables.
Table
24. The level
of pupils'
errors
of problem
solving
in urban
and rural
F r e q u e n c y o f f a ilu r e o n P r o b le m s o lv in g le v e l
I . R e a d in g
II . U n Cd eorns ta
c enp dt in g o f
III. P ro c e ss
o
z
o
c
<5j
pu
Q 5 (3 )
Q 6 ( l)
Q 8
S3
f-c
Pi
cS
-｣+-ｻ
c
nj
V S^ ^ H
DI10
6
10
OS
蝣
Qi
10
6
10
cS
蝣
i+-ｻ
S
20
12
20
19
c
ett
a
I^ F s ^ ^ H I t3
I
Pi
ID
10
2
6
9
8
8
schools
N o . o f s tu d e n t s
w it h C o r r e c t
A n sw e r
ed
4->
s
19
10
14
a
a
pu
ｫS
IH u= .^ ^ B
Pi
0
8
1
2
4
2
S3
4-"
｣蝣
1
10
6
International
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Project Towards the Endogenous Development of Mathematics
Education
Final Report
Table 25. The level of pupils' errors of problem
according to their performance
solving
Frequency of failure on Problem solving level
III. Process
II. Understanding
of
Concept
No. of students
with Correct
Answer
I. Reading
a
I
I
!
1
o
I
a.
o
Q5 (3)
Q6(l)
10
Q8
10
o
I
i
p.
I
10
10
20
12
20
CD
O
4)
I
=5
<L>
•E8
8.
a,
-a
O
«5
I
a.
•Ea
o
10
19
10
14
I
8,
I
o
10
T a b le 2 6 . P u p ils ' m is ta k e s in p ro b le m s o lv in g in Q 5 (3 )
Q 5 (3 )
D ifficu lt w o rd s
an d E x p re s sio n s
A ny
rem a rk s
re g ard in g
p ro b lem so lv in g
p ro c e ss
S p e c ifi c
m ista k e s
U rb a n sc h o o l
R u ra l S c h o o l
H ig h
p e rfo rm a n c e
" C o rre ct" ,
" C o rre ct" ,
" C o rre c t" ,
" ap p ro p riate ''
" a p p ro p r ia te''
" a p p ro p riate "
1. 1/4 is g rea ter 1 . 1 /4 is g re ate r 1 . 1/4 is g re a te r
th a n 1/3 , sin c e 4 th a n 1/3 , s in c e 4 th an 1 /3 , s in c e 4
is g re ater th a n 3 . is g re ate r th a n 3 . is g re a te r th a n 3 .
2 . A : 1/4 , B : 1/3 : 2 . A : 1/4 , B : 1 /3 : 2 . A : 1/4 , B : 1/3 :
A =B
B =A
A =B
1.
If
1.
If 1 .
If
d e n o m in a to r is d e n o m in a to r is d e n o m in ato r is
g re ate r
th an b ig
th en
th at g re a te r
th an
o th er
fra c tio n 's v a lu e o th e r
d e n o m in a to rs
d e n o m in a to rs
IS
th e n th a t fra ctio n
th en th at frac tio n 2 . D o n 't k n o w
h
o
w
to
i
f
n
d
o
u
t
is b ig g e r th an
is b ig g er th a n
th e so lu tio n .
o th er fr a c tio n s.
o th e rs .
2 . D o n 't k n o w
h o w to fi n d o u t,
h o w to fin d o u t,
w h ich is g re a te r
w h ic h is g re ate r
th a n o th e r.
th an o th e r.
20
Low
p e rfo rm a n c e
" C o rre ct" ,
" ap p ro p riate "
1 . 1 /4 is g re ate r
th a n 1/3 , sin ce 4
is g re a te r th an 3 .
2 . A : 1/4 , B : 1/3 :
B =A
1.
If
d e n o m in a to r is
b ig
th e n th a t
fra c tio n 's v a lu e
IS
h o w to fin d o u t
th e so lu tio n .
International
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Q 6 ( l)
U rb a n sc h o o l
D iffi c u lt w o rd s
an d
E x p re ssio n s
H ig h
p e rfo rm a n c e
" P ro c e s s"
"P ro c e ss"
A ny
re m a rk s
re g a rd in g
p ro b le m
s o lv in g p ro c e ss
1 . C a lc u latio n is
a c c u ra te
but
c o u ld
not
e x p la in
th e
p ro c e ss .
S p e c ifi c
m istak e s
1 . O n ly w ro te
th e a n sw e r th a t
w a s 5 /3 in ste a d
o f sh o w in g a n y
p ro c e s s.
Table 28. Pupils'
Q(8)
Urban school
Difficult
words
and
Expressions
Any remarks
regardin g
problem
solving process
"Relation ship" ,
"in terms of
Specific
mi stake s
1. Pupils have no
idea how to find
out
the
relationship.
1. Only wrote
the answer.
1 . A d d e d th e
n u m e ra to rs a n d
d e n o m in ato rs.
H e /s h e h a s n o
id e a
re g a rd in g
frac tio n .
2 . W ro te th e
p ro c e ss
but
c o u ld n o t fin d
o u t th e so lu tio n .
1 . C o u ld n o t fi n d
o u t th e
le a st
com m on
m u ltip le .
I 2 . C o u ld n o t
w rite th e frac tio n
co rre c tly .
mistakes
1 . A d d e d th e
n u m e ra to rs a n d
d e n o m in ato rs .
H e /sh e h a s n o
id e a
re g a rd in g
fra c tio n .
L ow
p e rfo rm a n c e
1. C a lc u la tio n is
a c c u ra te
but
c o u ld
not
e x p la in
th e
p ro c e ss .
2 . W ro te th e
p ro c e ss
b ut
c o u ld n o t fi n d
o u t th e so lu tio n .
1 . C o u ld n o t fi n d 1 . O n ly w ro te
o u t th e
le a st th e a n sw e r th at
w a s 5 /3 in ste ad
com m on
m u ltip le .
o f sh o w in g a n y
in problem
p ro ce s s.
2 . C o u ld
not
w rite th e fr a ctio n
c o rre c tly .
solving
in Q (8)
High
Low
p erformance
performance
"Bracket"
"Relationship", "Bracket"
"in terms of,
"Bracket"
1. Only wrote
1. Only wrote
the answer that
the answer that
was5/3.
was5/3.
2. Only wrote
2. Only wrote
the answer.
the answer.
1. Pupils have no 1. Pupils have no 1. Pupils have no
idea how to find idea how to find idea how to find
out
the out
the out
the
relationship.
relationship.
relation ship.
Rural School
2.1.5 Discussion
2.1.5.1
Discussion
from the First Year Field
Survey
In general, the age of students in grade 4 ranges from8 to 10 years. In the case of Bangladesh
(Table 7), it varies from 8 to 14 years because of repetition of grade and tendency to begin
school at a later age, etc. Therefore, approximately 25% students are in the age group of ll to
14years.
21
International
Cooperation
Project
ofMathematics
Education
Final Report
According to Table 8, over 65% students possess approximately
0-10 books at home excluding
magazines, newspapers, and their schoolbooks
in both urban and rural schools. In comparison,
(Table 9) students in the urban school possess the following
items, with the exception of a
computer, in a greater number than students in the rural school: Calculator,
TV, radio, study
desk, a quiet place to study, dictionary,
and supplementary
books. In the survey, students in the
rural school have reported possessing
a greater number of computers than those in the urban
school; however, it is assumed that the students in the rural school misunderstood
the concerned
item in the questionnaire.
Table 10 indicates
that generally
the
According to the findings,
the average
is 1.5, think learning mathematics will
mathematics is 1.3, and need to do well
students have a positive
attitude
toward mathematics.
percentage of students who enjoy learning mathematics
help them in their daily life is 1.4, usually do well in
in mathematics to get into the university
of their choice
is1.3.
Table 1 1 indicates
that students frequently
do a large number of mathematical
exercises such as
4 fundamental rules, fractions,
draw geometrical
shapes, etc. in their mathematics lessons and
relate what they are learning in mathematics to their daily lives.
From Table 12 it is apparent that students like their school and teachers. They believe that
teachers help students to improve their performance and the students themselves also attempt to
do their best in this regard.
Table 13 indicates
that students spend some time before or after school to watch TV, play or talk
with friends, do ajob at home, do homework for class, etc. It is impressive
that certain students
have reported reading books for enjoyment.
Although
the test achievement
results of the students
is not at the expected
level, the
performance of students in the urban school (correct answer 36.8%) is better than that of
students in the rural school (correct answer 29.1%). The performance difference
between the
two groups is 7.7%. Table 15 indicates
that the performance of both boys and girls in the urban
school is also better than that of the rural school. The performance difference
between the two
groups is 6.9% and 9.6%, respectively.
According to the analysis of content domain, the performance of students in the multiple-choice
test is better than in the short-answer test. In this respect, teachers stated that students are
unfamiliar
with answering written problems.
In general, the performance of students is
comparatively
good in data, number, and measurement content and poor in geometry and
algebra
content. It appears that teachers
lay a greater emphasis
on data, number, and
measurement content, consequently
causing students to perform better in these areas.
The performance of students in the urban school in content domain is better
in the rural school, except in measurement content in the short-answer test
performed better. The underlying
assumption of this exception appears to
in the measurement content are closely related to the rural context, thereby
in the rural school to have a better understanding
than students in the urban
than that of students
where the latter have
be that certain items
enabling the students
school.
The most difficult
question for students in the urban school in the multiple-choice
test was Ql-3
and the rate of correct answers is 4.3%; in the short-answer test were Q2-2, Q4-2, Q4-7c, Q6-2,
Q6-8, Q6-9, and Q6-12 and the rate of correct answers is 0%. On the other hand, the most
difficult
question for students in the rural school in the multiple-choice
test was Q3-12 and the
rate of correct answers is 0%; in the short-answer test the most difficult
were Q2-8, Q2-9, Q4-7c,
Q6-2, Q6-8, Q6-9, Q6-10, and Q6-12 and the rate of correct answers is 0%. According to the
teachers,
some of the abovementioned
questions
were out of syllabus
and some were too
difficult
for grade-4 students in Bangladesh.
22
International
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Further,
the questionnaire
reveals that although
the general attitude
mathematics is positive,
it is not reflected
in the result of their achievement
of students
test.
toward
Teachers argued that students are not accustomed to this type of questionnaire
format and
sentence problems. They also stated that certain items in the test were not covered in the
syllabus,
such as Ql-9, Q3-2, Q3-12, and Q6-9. å
Although additional
explanation
was provided
the students' understanding
of the questionnaire
2.1.5.2
Discussion
(a) Analysis
from the Second
of the Results
during the examination,
and test.
Year Field
it was not sufficient
for
Survey
of the Students'Achievement
Test
Table 22 presents the overall result of the students' score in the achievement test; this result was
not at the expected level. The average score is 42.9%. The performance of students in the urban
school (average score 5 1.7%) is better than that of students in the rural school (average score
33.7%). The performance difference between the two groups is 1 8%. It is evident that both boys
and girls from the urban school performed better than those in the rural school. The performance
difference between the urban and rural school is 21.5 for boys and 16.3 for girls.
The question-wise
analysis
of the achievement
of students
indicates
that, in general, the
performance of students is comparatively
better in Q4 (1), Ql (2), Q3 (1), and Q6 (1). It appears
that to a certain extent students are familiar with these types of questions, which are similar to
those in the classroom test. In general, the most difficult problems were Q6 (3), Q4 (2) Q10 Q5
(2), Q5 (3), and Q5 (l).
The most difficult
questions for students in the urban school were Q4 (2), Q6 (3), and Q5 (2)
and the average score is 0%, 0%, and 2.1%, respectively.
On the other hand, the most difficult
question for students in the rural school were Q2, Q10, Q 6 (3), and Q 5 (1) and the average
score is 0%, 8.6%, 13%, and 13%, respectively.
According to the teachers, the patterns of the
abovementioned questions are not the same as those of the textbook questions or classroom tests.
Teachers also reported that students are not accustomed to this type of questionnaire.
The performance of students in the rural school is comparatively
better in Q4 (2) and Q6 (3)
than that of those in the urban school. However, the reasons behind this are unclear. It is
possible
that students
in rural areas help their parents in measurement in their agricultural
activities;
thus, they are familiar with these items.
On the other hand, the performance of students in the urban school is better in Q2 than that of
those in the rural school. The reasons behind this, too, are unclear. It is possible
that students in
the rural school lack conceptual
understanding
of this topic, which is reflected in their inability
to relate division and fractions.
(b) Analysis
of Results
from the Interview
Interview items for students
understanding
of the concept,
are divided
(c) process,
into the following
four categories:
and (d) specific errors (if any).
(a) reading,
(b)
Table 24 shows the errors of students in problem-solving
in urban and rural schools. According
to the findings,
all students were able to understand Q5 (3), Q6 (1), and Q8; however, none of
them could understand the concept ofQ5 (3) and Q8 and only a few were able to understand the
concept ofQ6 (1).
The process skill:
A majority
of the students
were unable to show the correct process
23
of problem-solving.
In Q6
International
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FinalReport
(1) and Q8, halfofthe
students were unable to show the correct process. Further,
urban school were able to solve more problems than those in the rural school.
students
in the
Table 25 shows the level of errors in problem-solving
according to the performance of the
students. Students in all levels were able to understand Q5 (3), Q6 (1), and Q8; however, none
of them were able to understand the concept ofQ5 (3) and Q8, and few were able to understand
the concept ofQ6 (1).
In terms of the process skill, high performers are slightly
better than low performers, with the
exception of Q8 where the latter have performed better. High-performance
students were able to
solve a greater number of problems as compared with low-performance students, except in Q8.
In Q5 (3), the students of both
"appropriate".
They committed
1/4 is greater than 1/3, since 4
greater than other denominators
schools found it difficult to understand the words "correct" and
some errors in the process skill. For example, according to them,
is greater than 3 and again some stated that if a denominator is
then that fraction is greater than other fractions, etc.
In Q6 (1), the students of the rural school found it difficult
to understand the word "process."
They committed some mistakes in the process skill. For example, they added the numerators
and denominators in order to find the solution.
In Q8, students of both schools found it difficult
to understand the words "relationship,"
"in
terms of," and "bracket." They committed some mistakes in the process skill. For example, they
only wrote the answer, which was 5/3, instead of showing any process.
To conclude, it appears that lack of sufficient orientation
to various types of problems, the
so-called textbook-dependent
lesson plan, the tendency of copying problems from the textbook
for tests, and lack of adequate stimulation
for innovative thinking
are the main obstructions
for
students to achieve a satisfactory
level of understanding
in mathematics.
References
Directorate
of
Bangladesh-2001,
Primary
Ministry
Education,
DPE (2002).
Primary
of Primary and Mass Education Division,
JICA-Bangladesh
Project
Report (2005).
The Third
International
Cooperation Agency, Bangladesh.
The National Academy for Primary
primary school on the learning
Mymensingh, Bangladesh.
Pre-Evaluation
Education, NAPE (2002).
achievement
of children
24
Education
GOB.
Statistics
Study Report,
in
Japan
To study the impact of one shift
and management of school,
International
Cooperation
Project
Education
Final
2.2
Report
China
KANG BIAO JIN
HIDEKI IWASAKI
DAI QIN
Hiroshima
Hiroshima
Inner Mongolia
2.2.1
University
School
University
Normal University
Curriculum
-Mathematics education in China has witnessed drastic changes in the last decade. First, the
policy of "One curriculum many books" was introduced.
Absorbing
foreign experiences,
particularly
that of Japan, the government designs the curriculum and private corporations
are
entrusted
to write books for the same. As a result, there presently
are nine textbook
companies, although
during the 1990s there was only one book that used to be published
by
the People Education Press.
-In July 2001, the new curriculum "Math Curriculum Standard (Wc^l mmmwyreplaced the
previous curriculum "Math Education Curriculum (f^îSt^lfcffi).
" It will be tested until
201 0, and thereafter, it will be implemented
in the whole country.
-Among the 1st year samples, the urban schools used books based on "Math Curriculum
Standard,"
and the rural schools used those based on "Math Education
Curriculum."
With
regard to the 2nd year samples, both the urban and the rural schools used the latter.
2.2.2
Basic Information
(1)
Schooling
Age
Since the promulgation
of the "Compulsory Education Law of the People's Republic
of China"
in 1986, primary education extends over five or six years; junior secondary school, three or four
years, with a combination of6 + 3 or 5 + 4. In both cases, at present, nine years of schooling
in
primary and junior secondary schools is prevalent
in the current education system in China, and
almost all of the districts
opt for the 6 + 3 education system. Further, the entry-level
age for
primary schools is six or seven years; junior secondary schools, twelve or thirteen years.
(2)
School
Calendar
and Examination
In China, the academic year
the Chinese New Year (a day
other commences after the
schools have adopted a 5-day
includes two semesters: one from the beginning
of September to
between the end of January and February; it varies every year); the
Chinese New Year and extends till July. Moreover, since 1995,
week.
T a b le 1 . S c h o o l c a le n d a r in (y e a r )
Jan .
F eb .
蝣
M ar.
A p r il
M ay
S W iS Sl
Ju n e
Ju ly
P it
A ug.
S ep .
O ct.
N o v.
rｻ II ir
1
D ec .
gggjjggffi fHfttfgqKtfip
In China, schools adopt the two-semester system: the fall semester (Sep-Jan)
and the spring
semester (Feb-Jul);
they have introduced a 5-day week since 1995. The system of compulsory
education
is set at six years of elementary
school and three years of junior high school.
Although they are intended for children aged between six and twelve years, some elementary
schools are allowed to accept 7-year-old-children
as Grade 1 pupils.
25
International
Cooperation
Project
of Mathematics
Education
Final
(3)
Report
In China, there are fifty-five ethnic minorities
besides the Hans, which are in a majority. The
Chinese government has adopted an education policy in which each of them is supposed to use
both Mandarin Chinese (the language of the Hans) and their own language in school.
The 1st year sample schools are located around Beijing,
and the 2nd year sample schools, are
around Hohhot (the capital of Inner Mongolia).
The inhabitants
of both these areas use the Han
language (Chinese)
in both urban and rural areas.
(4)
Class
Organization
Daily classes commence at 8 am and end at 5 pm with a two hour lunch break in between.
duration of each lesson is forty-five minutes.
The
T a b le 2 . S c h o o l h o u r s
H o ur
8
9
10
ll
12
13
14
15
G ra d e
17
!:v* S 'S i;
2.2.3
Results
from the First
(1 )
Survey
Schedule
Year Field
Table
D a te
1 st N o v .
2 n d N ov.
Survey
3. Schedule
of data collection
A c tiv ity
C A 154
a rriv a l at B e ij in g
M e etin g o n re se a rch P ro f . IW A S A K I, M r. L I J IA N Z H O N G a n d M r. JIN
Q R e c o n fi rm in g re se a rc h sc h e d u le ｩ R e w ard in g
C o lle c tin g m ate ria ls o n m ath e m atic s to fi n d o u t th e p re se n t situ atio n o f
C h in e se m a th e m a tic s e d u c atio n ; " T e a ch in g P la n fo r P rim ary E d u c a tio n
(2 0 0 4 )," a n d " C u rric u lu m o f M ath e m atic s fo r P rim a ry e d u c a tio n " e tc .
V isit P rim a ry S c h o o l A (at B e ij in g ) a n d re se a rc h G ra d e 4 p u p ils
1 0 :0 0 T e st p art I , l l :0 0 T e st p a rt II ,
1 1 :5 0 R e c o rd in g c la sse s(F ra ctio n )
V isit P rim ary S c h o o l B (at th e p ro v in c e o f H e b e i) a n d re se a rch G ra d e 4 p u p ils .
9 :0 0 T e st P a rt I , 10 :0 0 T e st P a rt i
1 1 :0 0 D isc u s sio n , 1 2 :0 0 L u n c h
14 :0 0 - 1 5 :0 0 V is it P rim a ry S ch o o l attac h e d to B a o d in g n o rm a l sc h o o l
3 rd N o v
4 th N o v
5 th N o v .
6 th N o v
D isc u ssio n a t C h in a N a tio n a l In stitu te fo r E d u ca tio n a l R e se a rc h
Target Schools and Samples
(2)
^
16
^
^
^
^
^
^
U rban P rim ary Scho ol
R ural Prim ary School
Urban Primary School:
_
Table 4. Location of schools
= j
S chool location
X u an w u D istrict, B eijin g
Q ingy u an C ou nty
Founded in 1957, this School has 14 classrooms in two of its buildings,
26
International
Cooperation
Project
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Final Report
which have special
classrooms
for music, art, and computers;
all classrooms
have modern
teaching
facilities.
Further, the school has a 1,200 square meter playground
and has also been
designated
as a "key export-oriented
school" by the Xuanwu District.
Rural Primary School: The population
of the village
is 4,862;
the number of pupils is 362; the
enrollment
rate of pupils is 100%. There are 35 teachers in total, of which 6 are mathematics
teachers.
T a b le 5 . D is trib u tio n o f s a m p le
G rad e
M ale
F em a le
U rb an P rim a ry S ch o o l
4
39
40
R ura l P rim ary S ch o o l
4
33
38
蝣
B I^ ^ H
T ota l
72
78
T o tal
79
71
15 0
T a b le 6 . D istr ib u tio n o f stu d e n ts ' a g e
A ge
(y e ars)
8
9
10
ll
12
13
14
15
(3)
¥^
i^
^
^ m m
U rb an
2
47
29
1
S ^
N o . o f P u p ils
R u ra l
1
18
46
4
2
1^
T o tal
2
48
47
47
4
= 2^ =
1
Results of Student Questionnaire
ofstudents'
books (%)
0 -10
l l -2 5
2 6 -10 0
10 0 -
books
books
books
b ooks
C o m b in e d
2 2 .8 2
3 4 .9 0
3 1 .5 4
1 0 .7 3
U rb a n s c h o o l
1 0 .2 6
3 4 .6 2
3 9 .7 4
1 5 .3 8
R u r a l sc h o o l
3 6 .6 2
3 5 .2 1
2 2 .5 3
5 .6 3
T a b l e 8 . D is t r ib u t io n o f s t u d e n t s ' h o m e it e m s ( Y e< s , % )
C a lc u la to r
T V
R a d io
D e sk
Q u ie t
D ic tio n a ry
p la c e
C o m b in e d
6 4 .7
8 2 .3
8 0 .1
9 4 .9
5 4 .7
8 1 .0
6 8 .7
8 4 .8
27
6 3 .3
8 7 .3
74
9 2 .4
B ooks
(E x c lu d in
g
te x t b o o k )
68
8 7 .3
C o m p u te r
4 0 .7
6 4 .5
International
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Table 9. Students'
attitude
towards
mathematics
A v e ra g e (% )
P u p ils ' P e rc e p tio n s
C o m b in e d
U rb a n
R u ral
1 . 1 u s u a lly d o w e ll in m a th e m a tic s
1 .7
1 .5
1 .8
2 . 1 w o u ld lik e to d o m o re m a th e m a tic s in sc h o o l
1 .6
1 .4
1 .7
3 .1
3 .3
2 .8
3 . M a th e m a tic s is h a r d e r fo r m e th a n fo r m a n y o f
m y c la s s m a te s
4 . 1 e n jo y le a r n in g m a th e m a tic s
1 .4
1 .3
1 .5
5 . 1 a m j u st n o t g o o d a t m a th e m a tic s
3 .2
3 .3
3 .0
6 . 1 le a rn th in g s q u ic k ly in m a th e m a t ic s
1 .8
1 .6
1 .9
2 .8
3 .0
2 .6
2 .6
2 .7
2 .5
3
3 .1
2 .9
2 .4
2 .5
2 .3
7 . I th in k le a rn in g m a th e m a tic s w ill h e lp m e in m y
d a ily life
8 . 1 n e e d m a th e m a tic s to le a r n o th e r s c h o o l su b j e c ts
9 . I n e e d to d o w e ll in m a th e m a t ic s to g e t in to th e
u n iv e r sity o f m y c h o ic e
10 . 1 n e e d to d o w e ll in m a th e m a t ic s to g e t th e j o b I
w an t
Table 10. Students'
activities
in their
mathematics
lessons
A v e ra g e
S tu d e n t s ' a c tiv itie s
C o m b in e d
U rb a n
R u ra l
2 .8
3 .0
2 .6
2 .5
2 .7
2 .5
2 .9
3 .1
2 .9
2 .9
3 .0
2 .9
2 .4
2 .5
2 .3
1 .9
1 .8
2 .1
7 . 1 w o r k w ith o th e r s tu d e n ts in s m a ll g ro u p s
2 .1
2 .1
2 .2
8 . 1 e x p la in m y a n s w e rs
2 .3
2 .2
2 .3
9 . 1 lis te n to th e te a c h e r ta lk
1 .2
1 .2
1 .3
1 .8
1 .6
2
1 1 . 1 r e v ie w
2 .1
1 .8
2 .3
1 2 . 1 h a v e a q u iz o r te s t
2 .4
2 .2
2 .6
3 .3
3 .9
2 .6
1 . I p ra c tic e a d d in g , s u b tr a c tin g , m u ltip ly in g , a n d d iv id in g
w ith o u t u s in g a c a lc u la to r
2 . 1 w o rk o n fr a c tio n s a n d d e c im a ls
3 . 1 m e a su r e th in g s in th e c la s s ro o m
a n d a ro u n d th e s c h o o l
4 . 1 m a k e ta b le s , c h a r ts , o r g r a p h s
5 . I le a rn
a b o u t sh ap e s
su ch
as
c ir c le s , tr ia n g le s , a n d
r e c ta n g le s
6 . I r e la te w h a t I a m
le a r n in g in m a th e m a tic s t o m y d a ily
liv e s
m y h o m e w o rk
Table ll. Students'
attitude
towards
school and teacher
P u p ils' p ercep tio n s
1. 1 lik e b e in g in sch o o l
2 . I th in k th at stu d e n ts in m y sch o o l try to d o
th eir b est
3 . I th in k th at te ach e rs in m y sc h o o l care ab ou t
th e stu d en ts
4 .1 th ink th a t te a ch ers in m y sch o o l w an t stu d en ts
to d o th eir b est
28
C o m b in ed
A v erag e
U rb an
1 .7
1.5
R u ra l
1 .9
1 .6
1.5
1 .6
1.2
1.1
1.2
1.2
1.1
1.3
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Table 12. Students'
N o
activities
before
L e s s th a n 1 h o u r
or after school (%)
1 -2 h o u r s
tim e
1 . 1 w a tc h T V
M
o re th a n 2 b u t
le s s t h a n 4 h o u r s
4 o r m ore
h o urs
2 4 .8
4 6 .9
2 2 .1
4 .8
1 .4
2 . 1 p l a y o r t a lk w it h fr i e n d s
2 1 .8
3 5 .2 .
2 3 .9
l l .3
7 .8
3 . 1 d o jo b s at h o m e
l l .4
4 5
2 5
1 4 .3
4 .3
4 . 1 p l a y s p o rt s
1 2 .1
3 6 .4
3 7 .9
7 .1
6 .4
8 .0
3 1 .4
3 2 .1
2 1 .1
7 .3
6 . 1 u s e c o m p u te r
4 6 .4
3 7 .1
1 2 .9
0 .7
2 .9
7 .1 d o h o m ew o rk
1 .3 8
3 5 .9
3 3 .8
2 0 .7
8 .3
5 . 1 re a d a b o o k fo r
enjo y m en t
of students'
F a m ily m e m b e r
Results
member
C o m b in e d
U rb a n
2 -3 p e r s o n s
3 9 .7
6 1 .0
1 6 .0
4 -6 p e r s o n s
4 6 .6
3 5 .1
5 9 .4
7 -9 p e r s o n s
1 0 .3
2 .6
1 8 .8
10 -12 p erso n s
3 .4
1 .3
5 .8
0
0
0
right
answer (%
M o re th a n 1 2 p e r so n s
(4)
family
P u p ils ' D istr ib u t io n (% )
of Achievement
R u ra l
Test
of students'
A v e ra g e
M a le
F e m a le
H ig h e s t
L o w e st
C o m b in e d
7 2 .4
7 0 .5
7 0 .6
9 8 .7
1 3 .1
U rb an sch o o l
7 8 .0
7 5 .7
8 1 .3
9 8 .7
1 7 .7
R u ra l s c h o o l
6 6 .1
6 7 .1
6 5 .3
9 8 .6
8 .5
Table 15. Average rates of correct
answers by content
M u ltip le - c h o ic e (% )
N u m b er
domain
S h o r t-a n s w e r (%
7 7 .6
7 3 .1
M e asu rem en t
7 0 .7
5 6 .0
A lg e b ra
7 8 .2
4 9 .3
D a ta
8 1 .7
5 7 .0
P a tte rn s , re la t io n s a n d fu n c tio n s
8 2 .3
6 0 .0
G e o m e try
6 7 .3
6 7 .6
)
T a b le 1 6 . A v e r a g e r a t e s o f c o r r e c t a n sw e r s o f c o n t e n t d o m a in
b y u r b a n a n d ru r a l s ch o o l
I^ H ^ H B U r b a n s c h o o 1
C o n te n t d o m a in s
T o ta l
M u ltip le
S h o rt
M u lt ip le
S h o rt
l ^^H I
I^ ^ ^ BTôVJta
^ i^
p
-c h o ic e
-a n s w e r
-an sw e r
- c7h4o .1
ic e
N u m b er
8 0 .8
7 9 .2
80
6 6 .3
7 0 .2
7 5 .8
6 0 .8
6 2 .8
6 4 .9
5 0 .7
5 7 .8
A lg e b r a
8 3 .9
5 1 .9
6 7 .9
7 1 .8
4 6 .5
5 9 .2
D a ta
8 8 .0
6 9 .3
7 8 .7
7 4 .7
4 3 .3
59
P a tte rn s , r e la t io n s
8 7 .1
5 7 .0
7 2 .1
7 6 .9
6 3 .4
7 0 .2
)
I^ ^ ^^ Î A S^ ^ ^ H l
I^^ ^ H A Sff ^ ^ ^ H 蝣^ ^ ^ ^ ^ ^ ^b ^ ^ ^ H 蝣^ ^ 蝣^ ^ ^ ^ 蝣1
a n d fu n c tio n s
29
International
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Education
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2.2.4
Results
from the Second Year Field
(1 )
Survey
Schedule
Table
Survey
17. Schedule
of data collection
A ctiv ity
C A 1 54 , (H iro sh im a) 14 :2 5 - (B e ijin g ) 17 :2 5
C A l 1 16 , (B eijin g ) 2 0 :2 0- 2 1:2 5 (M on go lia )
^ R e co n fi rm atio n o f th e sch e d u le an d activ ities.
D ate
1 8 th N o v em b er
1 9 th N o v e m b e r
A .M . : L ectu re b y P ro f. Iw asak i at th e N o rm a l sc h o o l.
P .M . : R o un d -ta b le ta lk s w ith th e m ath e m atics te ach ers .
R e search on th e 5th g ra d ers at th e urb an p rim ary sc h o o l.
R o u n d-tab le ta lk s w ith th e tea ch ers at th e sch o o l.
V id e o -sh o o tin g .
R esea rch o n th e 5 th g rad ers at th e ru ral p rim ary sch o o l.
R o u n d -tab le ta lk s w ith th e tea ch ers at th e sch o o l.
V id eo -sh o o tin g .
R o u n d -tab le talk s w ith th e g ra d u ate stu d en ts a t In n er
M o n g o lia N o rm al U n iv ersity .
L e ctu re at v o catio n a l n o rm a l sch o o l.
M eetin g fo r th e rese arch co n ten ts an d m ark in g .
2 1 st N ov e m b er
2 2n d N o v e m b e r
2 3 rd N o v em b er
2 4 th N o v em b er
T ra in trav el : 2 1 :12 - 7 :3 0 (n e xt m o rn in g ).
A rriv a l in B eijin g .
B ack to Jap an . C A 15 3 (B e ijin g ) 8 :3 0 - (H iro sh im a) 1 :3 0
2 5 th N o v e m b er
Target Schools
(2)
Report
and Samples
Table 18. Location
of schools
S c h o o l lo c ation
H u h eh ao te c ity in Inn er M o n g o lia
T h e su b u rb s o f Inn er M o n g o lia
U rb an P rim ary S ch o o l
Huhehaote city and the rural area that are situated in Inner Mongolia were selected on the basis
of our 1st year field survey in which Beijing and the rural area were selected. It is extremely
important to comprehend the basic achievements as well as the teachers' evaluation on
mathematical education in China, where there exist more regional differences than other
developing countries.
The selected urban and rural primary schools are situated in Inner Mongolia. The target students
included Hans as well as other minority groups; the Han language is used as the medium of
instruction.
The condition of the facilities
as well as the teachers' academic backgrounds in both schools
weremostly similar.
r^
^
^
B
i^
^
U rb an P rim a ry S ch o o l
R u ra l P rim ary S ch o o l
T o tal
H
i G rad e
5
5
B oys
28
28
56
30
of sample
G irls
31
20
51
A ge
l l.6 8
l l.9 8
l l .83
T o ta l
59
48
10 7
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(3)
Results
of Achievement
Test
Table 20. Results
of the achievement
test (%)
A v erag e
B oys
G ir ls
H ig h e s t
L o w e st
6 4 .9
65
6 4 .4
9 8 .3
6 .8
R u ral S c h o o l
5 9 .7
65
54
9 1 .2
4 .2
A ll
6 2 .3
65
5 9 .2
T a b le 2 1 . Q u e s t io n - w is e a c h ie v e m e n ts o f p u p ils in p e r c e n t a g e
I^ H H ^ ^ ^ M M U
B oys
G ir ls
C o m b in e d
B oy s
i)
9 8 .3
10 0
9 6 .8
8 9 .6
9 2 .9
85
(2 )
9 6 .6
9 6 .4
9 6 .8
8 7 .5
9 2 .9
85
Q 2
4 4 .1
3 9 .3
4 8 .4
3 3 .3
3 5 .7
30
Q 3 (D
8 6 .4
8 5 .7
8 7 .1
8 7 .5
8 5 .7
90
Q 3 (2 )
8 4 .7
8 5 .7
8 3 .9
8 7 .5
9 2 .3
80
Q 4 (l)
9 8 .3
9 6 .4
10 0
8 9 .6
9 6 .4
80
Q 4 (2 )
6 .8
7 .1
6 .5
4 .2
7 .1
0
Q 4 (3 )
1 3 .6
2 1 .4
6 .5
4 .2
7 .1
0
Q 5 (l)
5 4 .2
5 7 .1
5 1 .6
6 2 .5
6 4 .3
60
Q 5 (2 )
2 8 .8
3 5 .7
2 2 .6
1 0 .4
1 4 .3
5
Q 5 (3 )
6 4 .4
6 4 .3
6 4 .5
3 9 .5 8
4 6 .4
3 0
Q 6 ( l)
100
10 0
100
8 9 .6
9 2 .9
85
Q 6 (2 )
9 8 .3
10 0
9 6 .8
9 1 .2
9 6 .4
85
Q 6 (3 )
4 7 .5
4 2 .3
5 1 .6
5 4 .2
9 2 .9
50
Q U
(4)
S c h o o l in r u ra l a r e a
C o m b in e d
G ir ls
0 7
6 1 .0
6 4 .3
4 8 .4
50
6 0 .7
35
Q 8
3 8 .9 8
3 5 .7
4 1 .9
3 7 .5
3 2 .1
45
Q 9
7 1 .1
6 7 .9
7 4 .2
8 3 .3
8 9 .3
75
Q lO
7 4 .6
6 7 .9
8 0 .7
7 2 .9
7 1 .4
75
Results
of Interview
Table 22. The pupils'
using NewmanProcedure
errors of problem
solving
level in urban and rural schools
F r e q u e n c y o f f a ilu r e o n P ro b le m so lv in g le v e l
I . R e a d in g
II . U n Cd eornsta
c enp dt in g o f
III . P r o c e s s
N o . o f stu d e n t s w ith
C o rr e c t A n s w e r
o
｣
a
?5 (3 )
Q 6 ( l)
Q 8
c
C3
x>
DUi
Ok*S
3
Pi
efl
｣4-J
0
0
0
0
0
0
0
c
<J3
x>
DL-<
C3
｣L_i
Kt
+-ｻ
e2
0
0
31
a
03
.｣J-.
>
D
C3
t3
Pi
2
0
8
4
1
10
a
4-*
e2
6
1
18
a
Xat>
DLh
8
10
2
CO
i-1
3
Pi
6
9
0
cs
｣-*->
14
19
2
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errors of problem
solving
F r e q u e n c y o f f a il u r e o n P r o b le m
I I . U n Cd eo rns tc ae np dt in g o f
I . R e a d in g
o
z
o
<u
o
<a
o
5i
s・a
3
<D
o
l
・
a
C3
｣+ ->
1 >- (
Ld h>
o.
0>
u
a
5
Lh
u
a .
a>
&<
Q 5 (3 )
0
0
0
Q 6 (l)
0
0
0
Q 8
0
0
0
IB
S
B
U rb an sch o o l
N o
D id n ot le arn
to
s o lv e
p ro b lem
in
d iag ram .
th in k s ab ou t
an d 1/3 .
・a
how
th e
a
H e
1/4
f-4
<D
a .
C o r re c t A n s w e r
K)
+ ->
t2
;S
DIu .
sx
*
C8
｣-(- >
<u
f-4
Q*
5
6
9
5
14
1
0
1
9
10
19
8
10
18
2
0
2
in problem
solving
in Q5 (3)
N o
H ig h p erfo rm an c e
N o
L o w p erfo rm an c e
N o
1/4 is sm a lle r th an
1/3 , b e cau se 1/4
h a s 4 p a rt s, b u t 1/3
h a s o n ly 3 p arts.
1/4 is sm a ller th an
1/3 , b e ca u se 1/4
h a s 4 p arts, b ut 1/3
h a s o n ly 3 p a rts.
T h e d en o m in ato r
o f A is g re ater th an
th at o f B , a n d th e
nu m era tor o f A is
sam e a s th at o f B .
S o A is g reater
th an B . H o w ev er,
sh e c an n o t g et th e
co rre ct an sw e r b y
d raw in g a fi g u re.
mistakes
in problem
solving
H igh p erfo rm a n c e
N o
T y p ica lly, d id n ot
u se a d iag ram to
so lv e th is k in d o f
T y p ic ally , d id n o t
u se a d iag ram to
so lv e th is k in d o f
p rob lem , so w a s
u n ab le to d o it,
ev en
th o u g h
te ach er h a d tau g h t
th is
m eth o d
b efo re.
N o
p ro b lem , so w a s
u n ab le to d o it,
ev e n
th o u gh
tea ch er h ad tau g h t
th is
m eth o d
b e fo re .
Sp ec ific m istak es
32
i
in Q6 (1)
R ura l S ch o o l
U rb an sc h o o l
N o
<L>
u
a
i
u
D<
i^ ^ ^^ ^ ^^ ^ ^^ ^ ^^ ^ ^^ m
Table 25. Pupils'
<u
o
a
s
1
S p e cifi c m ista k e s
Q 6 (l)
D iffi cu lt w o rd s &
E x p ressio n s
A ny re m a rk s
re g ard in g p rob le m
so lv in g p ro ce ss
1>
o
a
I
I
J-ID .
o .
mistakes
performance
N o . o f s t u d e n t s w i th
<L>
o
es
｣+ ->
to their
II I. P ro c e s s
!
Table 24. Pupils'
Q 5 (3 )
D iffi c u lt w ord s &
E x p re ssio n s
A ny
rem ark s
reg ard in g
th e
p rob lem
so lv in g
p ro cess.
level according
s o lv in g l e v e l
N o
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T a b le 2 6 . P u p ils ' m is ta k e s in p ro b le m so lv in g in Q (8 )
Q (8)
D iffi c u lt w o rd s &
E x p re ssio n s
A n y rem ark s
reg ard in g p ro b lem
so lv in g p ro c ess
U rb a n sc h o o l
S p e cifi c m istak e s
2.2.5
R u ra l S ch o o l
N o
N o
H ig h p erfo rm an ce
N o
L o w p erform an ce
N o
M e at B is 3 /5 o f
m ea t A .
H o w m u ch is th e
to ta l w eig h t o f
m e ats A an d B ?
M e at B is 3 /5 o f
m ea t A .
H o w m u ch is th e
to tal w eig h t o f
m eats A a n d B ?
C a n n ot d raw a
d ia g ra m to so lv e
th is p ro b lem .
M e at A is 3 k g an d
m e at B is 5 k g , so
th e an sw er is 3/5 .
She
does
not
u n d ersta n d
th is
p rob le m .
M eat A is 3 k g an d
C a n n o t d raw a
d iag ram to so lv e
th is p ro b lem .
m eat B is 5 k g , so
th e a n sw er is 3/5 .
She
d o es
not
un d e rstan d
th is
p ro b lem .
Discussion
(1)
-Students'
experience
in writing
The speed of answering
exams
test 1 was slower than that of test 2.
The explanation
was inadequate. For example, when being asked about the number of family
members in a multiple-choice
question, many pupils selected the number, which corresponded
with the number of their actual family members. When queried about the kind of facilities
available
in their homes, some students chose to skip the question.
No. of books or possession
of items at home
According to Table 7, the highest ratio of possession
of items at home in the urban area was in
the range of26 to 100 books (39.74%),
but on the other hand, that of the rural area was in the
range of 0 to 10 books (36.62%).
The highest collective
ratio including
both areas combined
was in therange
ofll to 25 books.
Based on Table 8, it is evident that people in the urban area have more items than those in the
rural area. In general,
people in the rural and urban areas have a TV (Q.V-b; 80.67%),
a
dictionary
(Q.V-f; 74%) and a desk (Q.V-d; 68.67%).
-Students'
attitude
toward mathematics
Table 27. Students'
1
attitude
toward
mathematics
I lik e m a th e m a tic s
I
n eed
to
stu d y
10 2 (7 0 .8 % )
1 0 3 (7 2 .5 % )
2
3 1((2 1 .5 % )
2 4 ( 16 .9 % )
6 (4 .2 % )
1 0 (7 .0 % )
3
4
5 (3 .5 % )
5 (3 .5 % )
m ath e m atic s to a c q u ire
a j o b o f m y c h o ic e
I
n ee d
to
stu d y
8 2 (5 7 .7 % )
2 2 ( 15 .5 % )
1 7 ( l l .8 % )
2 1( 14 .9 % )
m ath e m a tic s to o b ta in
a d m iss io n
in to
u n iv e rsity o f m y c h o ic e
The percentage of pupils that liked mathematics was 70.8%; 72.5% of them thought that they
needed mathematics in their daily lives; 58% of them thought that mathematics
was important
for obtaining
admission
into a university.
These results indicate
that the Chinese
pupils'
33
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Final Report
consciousness
of mathematics
equally in both urban as well
mathematics.
The most difficult
3 1%. Teachers
lines. Fractions,
by the Chinese
fractions (Q.2-2
correct answers
is considerably
high. Further,
this consciousness
is present
as rural areas, suggesting
the general positive
attitude
toward
question for the pupils was Q.6-9, and the percentage
of correct answers was
explained
that it was because the pupils had not yet learned how to draw parallel
proportions,
and decimal numbers were also considered
to be difficult questions
pupils.
Further analysis
will be necessary with respect to the questions
on
& Q.3-5) in which the Chinese pupils did not score well. The percentages
of the
for these questions
are as follows.
Table 28. Percentages
of the correct
answers in Q 2-2 and Q3-5
Q .2 -2
Q .3 -5
U rb a n
3 5 (4 4 .3 % )
4 1 (5 1 .9 % )
R u ra l
6 (8 .4 % )
2 4 (3 3 .8 % )
A v e ra g e
4 1(2 7 .3 % )
6 5 (4 3 .3 % )
Regarding
the utilization
of learning
material
sets, it was found that all the pupils
in the urban
area possessed a bag of learning
materials
and used them frequently
in their class. On the other
hand, it appeared that pupils in the rural area used the same sets sparingly.
This fact appeared to
affect the pupils'
achievement
in some domains. For instance,
the pupils'
achievements
in
locations
and special relationships
displayed
the widest gap: 68%; between the pupils
from the
urban and the rural areas (urban area: 89%; rural area: 21%; total: 57%).
(2)
Sample pupils G5 had not learned
Concepts of Fractions" at G4.
fractions
to a great extent;
they had only learned
the "Basic
The urban school had 1,405 pupils
including
280 from among the minorities.
There were 4
classes in G5 and 252 pupils.
On the other hand, the rural school had 877 pupils,
and 735 of
them were from other districts.
The percentage of the correct answer for Q2 was less than 50% and it meant that the concept
of fractions was unclear to the pupils. The performance for Q7 too was also poor.
Q10 required that asked the pupils frame a question. Most of the pupils framed it based on an
aspect of algebra;
few pupils attempted
to frame the question
by applying
the concepts of
quantity
in the fractions.
Here, we will analyze the results of the teacher's
responses to the questionnaire
them to describe the appropriate
method of teaching fractions.
From Table 5, it can
that both the urban and rural mathematics teachers wished to teach Q14, "Which is
or 1/3 m?" in terms of fractional
concepts. This implies
that the introduction
concepts in China is consistent
with the mathematics teachers' responses on Q14.
that required
be observed
longer 1/4 m
of fractional
The lowest percentages
of correct answers for the urban and rural areas were 6.8% and 4.2%,
respectively.
In response to Q4-2, fifty-two out of fifty-nine
students from the urban area
considered
1/2 m to be halfof2
meters. They had not yet learned mixed fractions at the time of
the survey.
The highest
percentages
of correct answers were for Ql(l),
Ql(2),
Q6(l),
and Q6(2).
For the
urban area: 98.3%, 96.6%, 100%, and 98.3%; for the rural area: 89.6%, 87.5%, 89.6% and
91.2%. It appears that the textbook
generally
containing
a great amount of arithmetic
and
34
International
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Final Report
problem solving
abovementioned
might have been the cause for these
questions.
higher
percentages
with respect
to the
Although the students faced problems in obtaining
the answer of Q1 0, 2/3, most of them used
algebraic
methods; few students faced problems in terms of fractional
concepts like quantity.
References
Chinese
Government (2000),
Day primary
"Mathematics
Education
Curriculum"(revised
shuppansha.
Chinese
Government
Standard"(pilot
version),
(2001),
Beijing
school
in Nine years' compulsory
education
from the pilot
version),
Jinmin-kyouiku
Day compulsory
education
Shihan-daigaku
shuppansha.
35
"Mathematics
Curriculum
International
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2.3
Ghana
HisashiKuwayama
ErnestKofiDavis
J. GharteyAmpiah
NarhFrancis
Hiroshima
Univeristy
of
Cape Coast
Univeristy of
Gape Coast
Mount Mary College
Educ ati on
2.3.1
University
School
of
Curriculum
In Ghana, the school mathematics
curriculum (syllabus)
is centralized.
According
to the
syllabus, the rationale behind teaching mathematics at the primary school level is to ensure that
mathematics at this level emphasizes knowledge and skills that will help pupils to develop the
foundation for numeracy. Pupils are expected to be able to read and use numbers competently,
reason logically,
solve problems efficiently,
and communicate mathematical
ideas effectively
to
other people. Their mathematical
knowledge, skills, and competence at this stage should enable
them to make more meaning of their world and should also develop their interest
in
mathematics as an essential tool for science, research, and development (MOE, 200 1).
The general
mathematics
aims of teaching mathematics at the primary school
syllabus for primary schools, include the following:
-develop basic ideas of quantity and shapes
-develop the skills of selecting and applying criteria
for classification
level,
as stated
in the 2001
and generalization
-communicate effectively
using mathematical
terms, symbols and explanations
through
logical reasoning.
-use mathematics in daily life by recognizing
and applying
appropriate
mathematical
problem-solving
strategies.
-understand the processes of measurement and acquire skills in using appropriate
measuring
instruments.
-develop the ability and willingness
to perform investigations
using various mathematical
ideas and operations.
work co-operatively
with other pupils to carry out activities
and projects in mathematics.
-see the study of mathematics as a means for developing
human values and personal qualities
such as diligence,
perseverance
confidence
and tolerance,
and as preparation
for
professions
and careers in science, technology,
commerce and a variety of work areas.
-develop interest
in studying
mathematics
The general objectives
of teaching
following:
At end of the instructional
-socialize
level
mathematics
at the primary school
period, each pupil will be able to
and cooperate with other pupils
-adjust and handle
to a higher
level
effectively
number words
-perform number operations
-makeuse of appropriate
strategies
of calculation
-recognize and use patterns, relationships
and sequence, and make generalizations
recognize
-use graphical
and use functions,
representations
formulae, equations
of equation
and inequalities
and inequalities
36
include
the
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Final Report
-identify/recognize
the arbitrary/
-use the arbitrary/
collect,
appropriate
process and interpret
standard
units of measures
unit to estimate
and measure various quantities
data
The primary school syllabus is designed to place significant
emphasis on the development and
use of basic mathematics knowledge and skills. The following
are the major areas of content
covered in all the six classes of primary schools:
-Number
Shape and Space
-Measurement
-Collecting
and handling
Data
-Problem Solving
-Investigation
with numbers
"Number" covers the reading and writing of numerals and the four operations
of addition,
subtraction,
multiplication,
and division.
"Investigation
with numbers" leads pupils to discover
patterns and relationships
and use the four operations meaningfully.
"Shape and space" includes
that which used to be known as "Geometry"; it is informally
introduced at the primary level by
using models and real objects in order to make the content more meaningful and interesting.
"Measurement" is intended to involve pupils in practical
activities
that will enable them to
appropriately
understand and use the various units. "Collecting
and handling
data" requires
pupils to be involved in the collection
of data from various sources and to learn to organize,
represent,
and interpret
the gathered
information.
"Problem solving"
is not considered
individually
in the syllabus. Problem solving activities
occur in nearly all the topics covered in
the syllabus as an essential aspect of emphasizing the practical application
of mathematics. It is
hoped that textbook developers will include appropriate
problems that will require mathematical
thinking rather than the mere recall and use of standard algorithms.
2.3.2
Basic
(1)
Schooling
Information
Age
Ghana presently
practices
the 6-3-3 system of preuniversity
education, namely, six years of
primary education,
three years of junior secondary education,
and three years of senior
secondary education (Educational
reforms, 1987). Pupils are supposed to begin school at the age
of six, even though their admission to schools depends on their parents' financial situation.
(2)
School
Calendar
and Examination
The school year begins in September and ends in June/July;
the school year is divided into three
terms, namely, the first, second, and third terms. The first term lasts from September to
December; the second term, from January to March; and the third term, from April to June/July.
Table
Ja n .
1. School
F eb .
Calendar
M a r.
A p ril
2 n d T e rm
(3)
Promotion
M ay
Ju n e
Ju ly
A ug.
Sep.
3 rd T erm
O ct.
N ov.
D e c.
1 st T erm
System
Promotions are conducted automatically
from primary schools to junior
37
secondary schools (JSS).
International
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Final Report
At the end of the third year in JSS, students take a national examination
(BECE: Basic
Education Certificate
Examination),
and the progression
to senior secondary schools (SSS),
technical
institutes,
or national vocational training institutes
depends on the result of the BECE.
(4)
Medium
of Instruction
The Ghanaian language is usually the medium of instruction
at the lower primary level (grades
one to three);
however, from the upper primary to the university
levels, the medium of
instruction
is English (Language policy, 1 973)
The controversial
issue regarding the medium of instruction
at the basic stages from grades one
to three has the attendant problem of poor achievements since the larger bulk of pupils are in
rural areas wherein English is far from being spoken or written on an everyday basis. At the
lower primary level wherein the mother tongue is accepted as the medium of instruction,
the
lack of relevant materials and difficulties
related to the ability of some teachers to communicate
in children's
native language-the
children
come from different
ethnic areas-have been
combined in order to render teaching at that level ineffective.
2.3.3
Results
(1)
Survey
from the FirstYear
Field
Survey
Schedule
Table 2 shows the schedule of our field survey. Since we decided to orally translate the test
items from English
to the local language, it took us considerable
time to administer
the
instrument. The implementation
of two tests on the same day seemed to be very tedious for the
pupils. Further, we had to use the mathematics periods provided in the class time table in order
to avoid a situation where teachers would be unable to teach certain subjects because of the
achievement tests. We, therefore, decided to use two days for the mathematics achievement test.
Weadministered
questionnaires
and conducted two achievement tests in a rural school between
February 16 and February 1 8 and in an urban school between February 21 and February 23. The
time duration for each test was also adjusted to provide sufficient time for the oral translation.
Table
2. Schedule
1 3 th /F e b /2 0 0 5
14 th
1 5 th
1 6 th
1 7 th
1 8 th
1 9 th
2 0 th
2 1 st
22n d
2 3 rd
2 4 th
(2)
Target
Schools
of the
A
M
N
Q
T
T
D
Q
P
T
T
D
First
Year Field
Survey
r r iv a l a t G h a n a
o v in g to C a p e C o a s t a n d M e e tin g
e c e s s a ry A rr a n g e m e n t s f o r th e f ie ld s u r v e y
u e s tio n n a ire in a ru r a l s c h o o l ( 6 0 m in )
e s t I (7 5 m in f o r P 4 , a n d 7 0 m in f o r P 5 )
e st II ( 7 0 m in f o r P 4 a n d P 5 )
ata In p u t
(D a ta In p u t)
u e st io n n a ir e in a n u r b a n s c h o o l ( 8 0 m in f o r P 4 , 4 5 m in f o r
5)
e s t I ( 7 0 m in fo r P 4 a n d P 5 )
e s t I I ( 6 5 m in f o r P 4 , a n d 7 0 m in fo r P 5 )
e p a rtu re f o r J a p a n
and Samples
Sample Procedure: Schools in Ghana are classified into three categories: schools A, B, and C.
These categorizations were performed based on the schools' performances in the performance
monitoring test (PMT) and the results of the BECE. Since this study focused on an "average
school," two basic schools categorized as school B, one each from urban and rural areas, were
38
International
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of Mathematics
Education
Final
Report
randomly selected from a list of schools B that was obtained from the Ghana Education Service
(GES) district
office for the purpose of the study. The sample comprised two average schools,
one each from urban and rural areas, in the Cape Coast municipal area in the central region of
Ghana. Both schools were noted to be average in terms offulfillment
of equipment.
The urban school was located in the town of Cape Coast and had a staff strength of nineteen
teachers, including
five KG teachers, six primary school teachers, six JSS teachers, and two
head teachers-one each for KG and primary/JSS-who
were detached. The school comprised
300 pupils. On the other hand, the rural school was located approximately
25 km north of Cape
Coast and had a staff strength offifteen teachers, including
two KG teachers, six primary school
teachers, six JSS teachers, and a head teacher who was detached. This school consisted of 180
pupils.
During the planning of our field survey in October 2004, the local team in Ghana recommended
that the test and questionnaires
be administered
to grade five pupils from the primary school,
since the pupils who were currently in grade four had been promoted to this grade barely a one
month ago. For this reason, the grade four pupils had not covered most of the topics included
in
the test to be administered.
In reality, the field survey was implemented for both grade four and
five pupils. In both the schools,
six classes were held at the primary school level with class
teaching
and three classes were held at the JSS level with subject teaching.
Both the schools
held one class each under grades four and five so that the pupils could be treated as samples.
Tables 3 and 4 present the basic information
regarding sample pupils.
Table
3. Distribution
of Sample
G rad e
4
5
4
5
U rb a n P r im a ry S c h o o l
M a le
17
13
16
16
62
T o ta l
Table
4. Distribution
of Pupils'
U r b a n P rim a ry S c h o o l
(3)
Results
of Student
Y o u n g est
O ld e s t
A v erag e
P 4
9
14
l l .2
P 5
P 4
P 5
10
9
10
16
14
14
l l .8
1 0 .9
l l .8
Questionnaire
of Pupils'
books
(%)
0 -10
l l -2 5
2 6 -10 0
M ore
b ooks
books
books
10 0 b o ok s
th a n
n .a
T o ta l
P 4
8 3 .3
1 2 .5
0
0
4 .2
10 0
P5
3 8 .5
4 6 .2
l l .5
3 .8
0
10 0
T o ta l
6 0 .0
3 0 .0
6 .0
2 .0
2 .0
10 0
P 4
1 3 .2
4 2 .1
3 4 .2
1 0 .5
0
10 0
P 5
25
4 6 .4
14 .3
14 .3
0
10 0
1 8 .2
4 3 .9
2 5 .8
12 .1
0
10 0
3 6 .2
3 7 .9
1 7 .2
7 .8
0 .9
10 0
R u ral
U rb a n
T o ta l
38
28
24
26
116
age
A g e ( y e a r s o ld )
F e m a le
2 1
15
8
10
54
T o ta l
C o m b in e d
39
International
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Table
6. Distribution
of Pupils'
Home Items (Yes, %)
C a lc u la to r
T V
R a d io
D e sk
D ic tio n a r y
C o m p u te r
R u ral
2 4 .0
7 6 .0
9 2 .0
8 4 .0
4 4 .0
1 2 .0
U rb a n
5 9 .1
8 4 .8
9 5 .5
5 9 .1
5 1 .5
1 6 .7
C o m b in e d
4 4 .0
8 1 .0
9 4 .0
6 9 .8
4 8 .3
14 .7
(Excludes
"no-response" and "impossible
Table 7. Pupils'
Attitude
to interpret")
Towards Mathematics
A v erag e
P u p ils ' P e r c e p tio n s
C o m b in e d
P4
P 5
1 .7
2 .0
1 .6
1 .3
2 .0
1 .2
1 .4
1.1
1 .3
1 .0
2 .2
2 .2
2 .4
2 .1
2 .1
2 . 1 w o u ld lik e to d o m o r e m a th e m a tic s in s c h o o l
th a n
fo r m a n y
of m y
c la s sm a te s
U rb a n
P 5
1 . 1 u s u a lly d o w e ll in m a th e m a tic s
3 . M a th e m a tic s is h a r d e r f o r m e
R u ral
P4
4 . 1 e n j o y le a rn in g m a th e m a t ic s
1 .3
1 .5
1 .3
1 .2
1 .4
5 . 1 am
2 .2
2 .2
1 .8
2 .1
2 .5
6 . 1 le a rn th in g s q u ic k ly in m a th e m a tic s
1 .8
1 .7
2 .0
1 .6
2 .0
7 . 1 th in k le a r n in g m a th e m a tic s w ill h e lp m e in m y d a ily lif e
1 .1
1 .2
1 .0
1 .1
1 .3
8 . 1 n e e d m a th e m a tic s to le a rn o th e r s c h o o l su b j e c t s
1 .4
1 .4
1 .6
1 .4
1 .4
1 .1
1 .2
1 .2
1 .1
1 .0
1.1
1 .2
1 .0
1 .2
1 .0
ju s t n o t g o o d a t m a th e m a tic s
9 . 1 n e e d to d o w e ll in m a th e m a tic s to g e t in t o th e u n iv e r s ity
o f m y c h o ic e
1 0 . 1 n e e d to d o w e ll in m a th e m a tic s to g e t th e j o b I w a n t
Excludes "no response" and "impossible
to interpret"
Legend: 1 (greatly
agree), 2 (slightly
agree), 3 (slightly
Table 8. Pupils'
Activities
in Their
Mathematics
disagree),
and 4 (greatly
disagree)
Lessons
A v erag e
P u p ils ' A c tiv itie s
C o m b in e d
U rb a n
R u ral
P4
P 5
P4
P 5
2 .0
2 .9
1 .1
2 .0
2 .3
2 .4
2 .4
2 .4
2 .3
2 .8
2 .4
2 .1
2 .9
2 .5
2 .0
4 . 1 m a k e ta b le s , c h a rt s, o r g r a p h s
2 .5
2 .2
3 .0
2 .3
2 .8
5 . 1 le a r n a b o u t s h a p e s su c h a s c ir c le s , tr ia n g le s , a n d r e c t a n g le s
2 .2
2 .1
2 .6
2 .5
1 .7
6 . 1 r e la te w h a t I a m
1 .6
1 .6
1 .6
2 .0
1 .3
2 .5
2 .0
2 .5
3 .2
2 .0
1 .7
2 .4
1 .8
1 .5
1 .2
9 . 1 lis te n to th e te a c h e r t a lk
2 .6
1 .3
2 .1
3 .2
3 .5
1 .5
1 .8
1 .8
1 .6
1 .0
l l . 1 r e v ie w m y h o m e w o r k
1 .5
1 .7
1 .6
1 .3
1 .4
1 2 . 1 h a v e a q u iz o r te st
2 .3
1 .9
2 .3
2 .3
2 .6
3 .8
3 .6
3 .6
3 .9
3 .9
1 . I p r a c tic e
a d d in g , su b tr a c tin g , m u ltip ly in g , a n d
d iv id in g
2 . 1 w o rk o n fr a c t io n s a n d d e c im a ls
3 . 1 m e a su r e th in g s in th e c la ss r o o m
a n d a r o u n d th e s c h o o l
le a r n in g in m a th e m a tic s to m y d a ily liv e s
7 . 1 w o r k w ith o th e r stu d e n t s in s m a ll g r o u p s
Excludes
Legend:
"no response" and "impossible
to interpret"
1 (every or almost every lesson),
2 (about
3 (some lessons),
and 4 (never)
40
half
of the lessons),
International
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Table
9. Pupils'
Attitude
Towards School
and Teacher
A v erag e
P u p ils ' P e rc e p tio n s
R u ral
C o m b in e d
U rb an
P4
P 5
P4
P 5
1 . 1 lik e b e in g in sc h o o l
1 .0
1 .0
1 .0
1 .0
1 .1
2 . 1 th in k th a t stu d e n ts in m y s c h o o l tr y to d o th e ir b e s t
1 .2
1 .4
1 .2
1 .3
1 .0
3 . 1 th in k th a t te a c h e r s in m y s c h o o l c a r e a b o u t th e s tu d e n ts
1 .1
1 .1
1 .3
1 .1
1 .0
1 .1
1 .3
1 .4
1 .0
1 .0
4 . I th in k th a t te a c h e r s in
m y
s c h o o l w a n t stu d e n t s to d o
th e ir b e st
Excludes "no response" and "impossible
to interpret"
Legend: 1 (greatly
agree), 2 (slightly
agree), 3 (slightly
T a b le
1 0 . P u p i l s ' A c t iv i t ie s
disagree),
and 4 (greatly
b e f o r e o r a f te r S c h o o l ( %
N o
L e s s th a n
1-2
M o r e th a n 2
t im e
1 h o ur
h o u rs
le s s t h a n 4 h o u r s
b u t
4
disagree)
)
o r m ore
h o u rs
n .a .
T o ta l
1 . 1 w a tc h T V
1 6 .4
4 7 .4
1 7 .2
6 .0
8 .6
4 .4
100
2 . 1 p l a y o r t a lk w it h fr ie n d s
9 .5
3 2 .8
2 9 .3
l l .2
1 4 .7
2 .5
10 0
6 .0
2 5 .9
3 1 .0
1 6 .4
1 6 .4
4 .3
10 0
4 . 1 p la y sp o r ts
8 .6
3 0 .2
2 3 .3
1 3 .8
1 9 .0
5 .1
10 0
5 . 1 r e a d a b o o k fo r e n j o y m e n t
5 .2
2 5
3 0 .2
2 2 .4
1 3 .8
3 .4
10 0
8 7 .1
5 .2
2 .6
2 .6
0 .9
1 .6
10 0
7 .1 d o h o m ew o rk
1 .7
4 1 .4
2 6 .7
8 .6
2 0 .7
0 .9
10 0
Table
Family
Member
6
7
8-
4 .2
8 .3
2 5 .0
l l .5
2 3 .1
2 6 .9
1 6 .0
2 6 .6
P 4
4 .2
4 .2
P 5
0
0
T o ta l
2 .0
2 .0
8 .0
Results
of Achievement
^
P4
P5
Table
of Pupils'
4
Scho o l
I^
Distribution
3
R u ra l
(4)
ll.
N o . o f p e o p le
^
H
P5
3 4 .6
3 .9
100
4 2 .0
3 .4
100
I
Score by Their
A v e ra g e
Grade and School
Location
SD
H ig h e s t
L o w e st
C o m b in e d
4 3 .6
9 .1 0
6 5 .8
2 1 .9
R u ra l
4 2 .5
1 0 .0 4
6 3 .0
2 1 .9
3 1 .5
U rb a n
4 4 .3
8 .5 1
6 5 .8
C o m b in e d
5 1 .4
1 0 .3 8
6 9 .9
15 .1
R u ra l
5 3 .7
8 .4 9
6 9 .9
3 5 .6
U rb a n
4 9 .2
l l .6 0
6 5 .8
15 .1
13. Distribution
P4
T o ta l
5 0 .0
n .a .
4 .1
100
Test
of Pupils'
^
5
of Pupils'
Score
by Their
Grade and Gender
A v e rag e
SD
H ig h e st
L o w e st
C o m b in e d
4 3 .6
9 .10
6 5 .8
2 1 .9
M a le
4 3 .1
8 .3 3
6 4 .4
2 6 .0
F e m a le
4 4 .3
10 .17
6 5 .8
2 1 .9
C o m b in e d
5 1.4
10 .3 8
6 9 .9
1 5 .1
M a le
5 4 .0
8 .2 7
6 9 .9
3 1 .5
F e m a le
4 8 .3
l l .8 2
6 7 .1
15 .1
41
(%)
(%)
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Table
14. Average
C o n te n t d o m a in
Rates of Correct
P4
Answers by Content
Domain (%)
P 5
M u ltip le - c h o ic e
S h o rt-a n s w e r
M u ltip le - c h o ic e
S h o rt- a n s w e r
N u m b er
4 2 .1
3 7 .9
5 8 .8
5 9 .3
N u m b e r ( F r a c t io n )
5 0 .2
2 4 .2
6 0 .8
4 0 .7
M e asurem e n t
5 1 .9
' 1 4 .5
4 5 .2
3 1 .5
G e o m e tr y
5 2 .0
4 0 .9
5 1 .4
4 6 .3
D a ta
6 3 .7
1 4 .1
4 8 .1
3 5 .2
P a tte rn s
5 4 .8
6 7 .7
7 0 .7
5 3 .7
A lg e b r a
4 3 .1
5 4 .8
4 9 .5
4 0 .7
Table 15. Average Rates of Correct
C o n t e n t D o m a in
N u m b er
N u m b e r (F ra c tio n )
M e a s u re m e n t
G e o m e try
D a ta
P a tte r n s
A lg e b ra
Table 16. Difference
C o n te n t D o m a in
N um b er
N u m b e r (F r a c tio n )
M e a s u re m e n t
Q u e s tio n T y p e
Answers by Content
G h a n a (P 4 , % )
T o ta l
U rb a n
M u ltip le C h o ic e
4 2 .1
S h o rt A n s w e r
3 7 .9
5 0 .2
2 4 .2
T o ta l
U rb a n
4 2 .8
4 1
5 8 .8
5 7 .7
5 9 .9
3 8 .6
3 6 .8
5 9 .3
5 7 .7
6 0 .9
4 6 .6
56
6 0 .8
5 6 .6
6 5 .4
2 1 .7
2 8 .1
4 0 .7
4 2 .9
3 8 .5
5 1 .9
5 9 .4
3 9 .9
4 5 .2
4 4 .9
4 5 .5
14 .5
1 9 .3
6 .9
3 1 .5
2 7 .4
3 5 .9
P a tte r n s
A lg e b ra
R ura l
52
4 8 .7
5 7 .3
5 1 .4
5 9 .8
4 7 .1
4 0 .9
39
4 3 .8
4 6 .3
4 4 .6
4 8 .1
6 3 .7
6 5 .8
6 0 .4
4 8 .1
5 5 .4
4 0 .4
14 .1
1 7 .1
9 .4
3 5 .2
3 2 .1
3 8 .5
5 4 .8
5 1 .6
60
7 0 .7
6 7 .1
7 4 .6
6 7 .7
7 1 .1
6 2 .5
5 3 .7
2 1 .4
8 .5
4 3 .1
4 0 .8
4 6 .9
4 9 .5
3 6 .6
6 3 .5
5 4 .8
50
6 2 .5
4 0 .7
0
8 4 .6
in Average Rates of Correct Answers by Content
between Grades and Locations
Q u e stio n T y p e
P5 - P4
T o ta l
U rb an
1 6 .7
S h o rt A n sw e r
2 1 .4
1 0 .6
S h o r t A n sw e r
1 6 .5
- 6 .7
S h o rt A n sw e r
1 .8
- 2 .2
2 4 .1
1 .8
-
10
9 .4
- 9 .4
2 1 .2
1 0 .4
- 6 .4
4 .4
5 .6
1 9 .5
- 0 .6
29
1 2 .4
- 8 .5
14 .5
5 .6
1 5 .6
1 5 .9
-
14
6 .4
-
1 8 .9
1 9 .1
l l .1
S h o rt A n sw e r
S h o rt A n sw er
1 4 .9
5 .4
P 5
- 0 .6
2 1 .1
1 4 .1
42
U rb a n - R u r a l
P 4
8 .1
-
S h o r t A n sw e r
-
Domain
R u ra l
17
S h o rt A n sw e r
D a ta
G h a n a (P 5 , % )
R u ral
G e o m e try
Domain in Each Location
-
3 .2
oQ .oQ
- 8 .6
1 2 .7
4 .3
- 4 .8
- 3 .5
1 0 .2
1 0 .4
- 2 0
5 .4
15
15
2 9 .1
7 .7
- 6 .4
1 5 .5
14 .6
- 8 .4
- 7 .5
- 4 9 .7
26
8 .6
- 6 7 .1
- 6 .1
- 2 6 .9
-
- 4 .2
1 6 .6
- 50
2 2 .1
-
1 2 .5
-
8 4 .6
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2.3.4
Results
from the Second Year Field
Survey
T a b le i 7 . S c h e d u le o f t h e S e c o n d Y e a r F ie ld S u r v e y
D a te
A c tiv ity
2 1 s t/ N o v /2 0 0 5
A r r iv a l a t A c c ra a n d M o v e to C a p e C o a s t ( J a p a n e s e R e s e a r c h e r )
22 n d
R e c o n fi r m a tio n o f th e S c h e d u le a n d A c t iv it ie s
M a th e m a t ic s A c h ie v e m e n t T e st o n 5 th G r a d e r s a t R u r a l P r im a ry
2 3 rd
S ch oo l
M a t h e m a t ic s A c h ie v e m e n t T e s t o n 5 th G r a d e r s a t U r b a n P r im a r y
2 4 th
S ch oo l
2 5 th
In te r v ie w t o P u p i ls a t U r b a n P r im a ry S c h o o l
2 6 th
D e p a rt u r e fo r J a p a n (J a p a n e s e R e s e a r c h e r )
2 7 th
N .A .
2 9 th
In t e rv ie w t o P u p ils a t R u r a l P r im a ry S c h o o l ( L o c a l T e a m )
(1)
For the second-year survey, we used the same sample pupils who were available for the schools
that were sampled earlier. The collected data were administered to grade five pupils from the
primary school, since they had just recently completed the grade four syllabus.
T a b le 1 8 . D is trib u tio n o f s a m p le
G rad e
5
5
U rb an S c h o o l
T o ta l
(2)
T o ta l
35
27
62
M a le
ll
19
30
F e m a le
24
8
32
A ge
l l .7 4
l l .6 7
l l .7 1
Results of Achievement Test
Table 19. Results
U rb an S c h o o l
A ll
of the Achievement
A v e ra g e
4 5 .3 7
4 2 .9 6
4 4 .3 2
M a le
F e m a le
5 2 .8 2
4 6 .4 2
4 8 .7 7
4 1 .9 6
3 4 .7 5
4 0 .1 6
Test (%)
H ig h e st
73
66
L o w e st
15
18
15
73
T a b le 2 0 . Q u e s t io n -W is e A c h ie v e m e n t s o f P u p ils ( % )
O v e r a ll
T o ta l
T o ta l
Q i (i )
8 7 .1
(2 )
8 4 .7
Q 2
R u ral S ch o o l
M a le
F e m a le
8 5 .8
10 0
90
9 5 .5
9 .7
17
3 7 .5
8 .5
0
0
0
Q 3 (1)
8 0 .9
8 5 .2
6 9 .7
9 2 .3
7 5 .3
8 2 .5
5 8 .3
Q 3 (2 )
7 7 .2
6 6 .7
8 4 .8
5 8 .3
9 0 .7
9 1 .2
8 9 .7
Q 4 (l)
7 9 .4
7 1 .4
9 0 .9
6 2 .5
8 9 .9
9 0 .7 5
8 7 .5
Q 4 (2 )
0 .8
1 .4
0
2 .1
0
0
0
0 4 (3 )
2 1 .0
2 7 .9
4 3 .1
2 0 .9
12
l l .9
12 .5
43
T o ta l
M a le
F e m a le
7 9 .3
89
9 4 .8
75
8 7 .5
7 7 .8
79
75
International
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Final Report
Q 5 (l)
5 0
4 0
3 6 .5
4 1 .5
6 3
6 8 .5
50
Q 5 (2 )
3 .2
5 .5
0
8 .5
0
0
0
Q 5 (3)
2 4 .2
2 8 .5
3 6 .5
2 5
1 8 .5
2 1
1 2 .5
Q 6 (l)
6 9 .2
7 2 .5
8 7 .5
6 5 .6
6 4 .9
6 4 .5
6 5 .6
Q 6 (2 )
5 6 .7
6 5
7 1 .6
6 2
4 5 .9
5 9 .9
1 2 .5
Q 6 (3 )
2 0 .0
2 6 .8
2 9 .5
2 5 .5
l l .1
9 .9
1 4 .1
Q 7
1 2 .9
1 4 .3
1 3 .8
1 4 .5
l l
1 3 .3
6 .2 5
Q 8
1 4 .5
l l .4
1 8 .1
8 .4
1 8 .5
2 6 .4
0
Q 9
5 3 .2
5 1 .5
6 3 .7
4 5 .8
5 5 .5
6 3 .2
3 7 .5
2 6 .6
2 7 .2
3 1 .8
2 5
2 6
3 1 .5
1 2 .5
Q
lO
Results
(3)
of Interview
using Newman Procedure
Errors
of Problem
F re q u e n c y o f f a ilu re o n P r o b lem
I . R e a d in g
a
Solving
Level in Urban and Rural Schools
s o lv in g le v e l
II . U n d e r s ta n d in g
E x p la n a t io n
N o . o f s tu d e n ts w ith
in E n g lis h
I II . P ro c e s s
o f Q u e stio n
o
z
・
nl
｣t-
a
+｣ -ｻ
d
(53
.pU t
D
C8
u*
｣
C3
+-t
e2
a
OS
.oJ-(
D
CIUiS
｣
C3
4->
｣蝣
4
5
9
7
6
13
3
1
3
0
0
0
0
1
1
5
8
9
8
17
9
9
18
1
c
(3
xii>
D
03
-3
・
I*
C8
e2
Q 5 (3 )
3
2
5
0 6 (1)
2
1
0 8
5
3
x >
DJ-
errors
Frequency of failure
of problem
solving
on Problem solving
KS
s
Pi
ca
+-ｻ
｣
4
4
2
6
4
9
10
9
19
2
3
2
2
4
level
by their
level
Exp lanation
II.
I. Reading
ｫa
x>
pJ-H
Understanding
of Problem
in English
III. Process
performance
No. of students
with Correct
Answer
8
•E8
«8
•Ea
Q5(3)
s
ex
I
•Ea
a,
I
8
I i•Ea
<u
e2
I
X
10
2.3.5
Discussion
According to Table 5, pupils
•E8
Oh
D.
o
åa
10
10
10
17
19
18
in an urban school possess
44
I
O
13
Q6(l)
Q8
•E8
o
more number of books at home than
International
Cooperation
Project
Development
of Mathematics
Education
Final
Table
23. Pupils'
Q 5 (3 )
D iffic u lt w ord s &
E xp ressio n s
S p e cific m istak e s
mistakes
in problem
solving
U rb an sch o o l
" ch o ic e"
" g re ater th a n"
- T h e p u p il ex p lain ed th e qu e stio n su ch
th at th e an sw e r is 1 w h en th e q u e stion
n u m b er is 1, an d th at th e an sw er is 2
w h en th e q u estio n n u m b er is 2 .
- S ho w ing that " _ + _ " in th eir w ritin g ,
4
3
an d ex p la in a w ron g p ro cess o f th e
ca lcu latio n su ch th at th e an sw er is 2
b ec au se 1+ 1= 2 in n u m erato rs.
- T h e p u p il h ap p en e d to p ick u p o n e
c h o ice an d c o u ld n o t ex p la in w h y .
- S h o w in g th e b ar d ia g ram w h ic h w a s
d iv id e d into th e w ro n g p o rtion s, an d
sa id " _ is g reater th an - "
4
3
Table
24. Pupils'
Q 5 (3 )
D iffi cu lt w o rd s &
E xp ressio n s
S p e cific m ista k e s
mistakes
by location
Report
in Q5 (3)
"ch o ic e"
"g reater th a n "
"b rack e t"
- T h e p u p il ex p la in e d th e q u e stio n su ch
th at th e an sw er is 1 w h en th e q u estion
n um b er is 1, a n d th at th e an sw e r is 2
w h en th e q u estio n n um b e r is 2 .
- S ho w ing th at " J_ + _ " in th e ir w ritin g ,
4
3
an d e x p la in a w ro n g p ro c ess o f th e
c alcu latio n su ch as {2 + 3 )ll .
- S h o w in g th e b ar d iag ram w h ic h w as
d iv id e d in to th e c o rrect p o rtio n s, b ut
sa id "I add ed J_
4 to _3 to g et 1 ."
- S h o w in g th e b ar d iag ram w h ich w a s
d iv id e d in to th e w ro n g p o rtio n s, a n d
sa id " i_
4 is greater than J3 _ ."
in problem
solving
by performance
in Q5 (3)
L o w p erfo rm a n ce
" ch o ic e"
" ch o ic e"
" g re ater th a n "
" b ra ck et"
- S ho w ing that " _L + _ " in th eir w ritin g ,
4
3
an d sa id "I d o n ot u n d erstan d th e
q u estio n ."
- S h o w in g th e b ar d ia g ram w h ich w a s
d iv id e d in to th e w ro n g p o rt ion s, a n d
- S ho w ing that " _ + _ " in th e ir w ritin g ,
4
3
an d ex p la in a w ro n g p ro ce ss o f th e
ca lcu latio n su ch th at th e an sw er is 2
b e cau se 1+ 1= 2 in n um era to rs.
- T h e stu d en ts ex p la in e d th e q u e stio n
su c h th a t th e an sw er is 1 w h en th e
sa id " _L4 is greater th an _3
qu estion n u m b er is 1, an d th a t th e
an sw er is 2 w h en th e qu estion n u m b er is
2.
- T h e p u p il h ap p e n ed to p ick up o n e
ch o ice a n d c o u ld n o t ex p lain w h y .
- S h o w in g th e b ar d iag ram w h ich w a s
d iv id e d in to th e c o rrect p o rtion s, b u t
sa id "I add ed ! to l to g et 1 ."
4
3
- S h o w in g th e b ar d iag ram w h ich w as
d iv id e d in to th e w ro n g p o rtio n s, a n d
sa id " _L is greater than _ ."
4
3
- S h o w in g th e b a r d iag ra m w h ich w a s
d iv id e d in to th e c o rre ct p ortion s, b u t
sa id " ! _ is g reater th an _ ."
4
3
45
International
Cooperation
Project
of Mathematics
Education
Final
Table
25. Pupils'
Q 6 (l)
D iff ic u lt w ord s &
E x pression s
mistakes
in problem
solving
by location
U rb an sch o o l
Report
in Q6 (1)
R u ra l S ch o o l
"C " a s litre
6 " as litre
"p ro ces s"
"co n ta in er"
p ro ce ss
"c o nta in er"
- A p u p il c o u ld n o t rea d th e q u estio n .
- A p u p il rea d S p ec ific m istak e s
Table
5
26. Pupils'
Q 6 (l)
D iffic u lt w o rd s &
E x p re ssio n s
as o n e fi v e, ｱ _ as tw o
fi v e
mistakes
- 1 + 2 = 13
5
5
5
in problem
solving
by performance
in Q6 (1)
A p up ilread _1 as o n e fiv e, io _ a s tw o
H ig h p e rform an ce
"p ro ce ss"
5
5
fiv e
" fl " as litre
p ro c ess
" c o n ta in e r"
S p ec ific m ista k e s
Table
Q (8 )
E x p re s sio n s
- 1 + 2 = 13
5
5
27. Pupils'
mistakes
in problem
solving
by location
U r b a n sc h o o l
in Q (8)
"Ju d y "
" r e la t io n s h ip "
" te r m s "
" w e ig h t"
" b o u g h t"
"Ju d y "
" r e la t io n s h ip "
" te r m s "
" w e ig h t"
" b o u g h t"
" G eo rg e"
" fr a c tio n "
" 5 k g o f m e a t"
- I t w a s d iff ic u lt fo r th e p u p il to
u n d e r sta n d th e w h o le q u e stio n .
" 3 k g o f m e a t"
S p e c ific m is ta k e s
- A p u p il S h e c o u ld n o t re a d th e q u e s tio n .
- " I d id n o t u n d e r sta n d th e q u e s tio n ."
- " I h a d to a d d th e tw o n u m b e r s , 5 k g a n d
3 k g ."
- " I s u b tra c te d 3 fr o m 5 to g e t 2 ."
- " I a d d e d 5 to 3 to g e t 8 , a n d th e n d iv id e d
8 b y 5 , b e c a u s e I d id th a t b e c a u s e th e
q u e st io n r e q u ir e d m e to e x p re s s th e
a n s w e r a s a fr a c tio n ."
- " I d id n o t u n d e r sta n d th e q u e stio n ."
- " I a d d e d G e o r g e 's m e a t to J u d y 's
m e a t ."
- " I s u b tra c te d 3 fr o m 5 to g e t 2 ."
- " T h e q u e s tio n w a n ts m e to m a k e 3 o r 5
1
a fr a c tio n . S o , th e fr a c t io n fo r 3 is 1 7 : :
Y o u h a v e to d iv id e it in to 2 to g e t its
fr a c tio n . S o , if 5 is to b e in fr a c tio n , it
1
w ill b e 2 -
46
International
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Project
of Mathematics
Education
Final
Table
28. Pupils'
Q (8 )
E x p r e s s io n s
mistakes
in problem
solving
H ig h p e r fo rm a n c e
in Q (8)
L o w p e rf o rm a n c e
" Ju d y "
" re la tio n sh ip "
" w e ig h t"
" b o u g h t"
S p e c ific m is ta k e s
by performance
Report
"Ju d y "
" r e la tio n s h ip "
" te r m s"
" w e ig h t"
" b o u g h t"
" G e o rg e "
- It w a s d if fi c u lt f o r th e p u p il to u n d e r sta n d
th e w h o le q u e s tio n .
- " I d id n o t u n d e r sta n d th e q u e st io n ."
- " I a d d e d G e o r g e 's m e a t to J u d y 's m e a t."
- " I s u b tr a c te d 3 fr o m 5 to g e t 2 ."
- " I s u b tr a c t e d 3 fr o m 5 to g e t 2 ."
- " I a d d e d 5 to 3 to g e t 8 , a n d th e n
d iv id e d 8 b y 5 , b e c a u s e I d id th a t
b e c a u s e th e q u e s tio n re q u ir e d m e to
e x p re s s th e a n sw e r a s a fra c tio n ."
- " T h e q u e s tio n w a n ts m e to m a k e 3 o r 5
1
a fr a c tio n . S o , th e fr a c t io n fo r 3 is 1
2.3.5
Discussion
(1 ) Discussion
Y o u h a v e to d iv id e it in to 2 to g e t its
fr a c t io n . S o , if 5 is to b e in fr a c tio n , it
about the first 1 year survey
e 2 According to Table w ill
5, b pupils
in an urban school possess more number of books at home than
those in a rural school. Besides, the rate of possession of certain items listed in Table 6, with the
exception of "desk and chair," suggests that pupils in an urban area are living in a better
material
condition.
It appears that such conditions
will reflect
on the pupils'
learning
environment at home. The result concerning "desk and chair" should be examined again in other
research attempts. Table 10 presents the result concerning pupils' activities
in a day, and Table
1 1 indicates
the result concerning the number of family members. These are also related to the
pupils'
learning environment.
The result provided in Table 7 shows that most pupils in Ghana have an affirmative
view with
regard to mathematics and their learning. Table 9 also shows the pupils'
positive
impressions
regarding
their schools and teachers. This may possibly
lead the pupils to maintain a positive
attitude
toward learning mathematics
even in the future. Some results, however, should be
critically
reexamined, since certain questions with negative sentences seem to be misunderstood
by pupils.
For example, the results do not appear to be significantly
different
despite the fact
that the sentence in item 5 ("I am just not good at mathematics")
appears to be almost contrary
to that in item 1 ("I usually do well in mathematics").
Although
Table 8 illustrates
that the choices
provided
in certain questions
may not be
appropriate,
the results obtained for items 8 ("I explain my idea") and 9 ("I listen to the teacher
talk") might be suggestive
of the way in which teaching is conducted in rural and urban areas.
Therefore, in order to countercheck this tendency, it is necessary to conduct an analysis of the
actual lessons conducted with regard to mathematics.
Table 12 provides
the overall results of the scores obtained by pupils in the achievement tests.
Since it appears to be a matter of course, the total score for grade five pupils was better than that
for grade four pupils.
With regard to some content domain, however, Table 14 reveals the
47
International
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of Mathematics
Education
Final
Report
results to be different.
In addition,
a contrary result can be found between the schools in the
rural and urban areas. Although the result obtained for grade four pupils in the rural school was
worse than that obtained for grade four pupils in the urban school, grade five pupils in the rural
school scored better than those in the urban school. Based on an interview with the head teacher
of the urban school, it was found that the class teacher for grade five had shown a relatively
lower performance in the urban school. This was because at the time the teacher was enrolled
for an upgrading course at Winneba University
without study leave and hence was unable to
give his pupils the undivided
attention that they needed.
In Table 13, gender is not found to be a consistent
tendency. Although
discussed
in mathematics
education,
further research
is required
investigation
in this regard.
a gender gap is often
to make a thorough
With regard to Table 14, we identify
some weak areas by grades and by question types. As
regards the multiple-choice
questions,
the following
are the weak areas for grade four pupils: (i)
Number, (ii) Algebra,
and (iii) Fraction.
Even though grade five pupils demonstrate improved
understanding
of "Number" and "Fraction,"
worse results are obtained
for other areas,
especially
"Measurement" and "Data." Better results are obtained for multiple-choice
questions
than for short-answer-type
questions.
Although
the pupils demonstrate
improvement in most
areas, their weak areas in short-answer-type
questions,
such as "Measurement," "Data," and
"Fraction,"
remain unchanged.
In Table 15, barring
some exceptions,
similar
results
are
observed in both rural and urban schools.
According to the curriculum
test match performed by collaborators
in Ghana, the following
topics covered in the test are not included in the grade four syllabus:
(i) subtraction
and addition
of decimals, (ii) ratio, (ii ) identification
ofa point in space, (iv) the concept of parallel
lines and
perimeters,
(v) identification
of surfaces, edges, vertices, etc., of solids, and (vi) transformations.
The lack of knowledge regarding these topics was partially
but definitely
reflected
in the pupils'
performances. It should be noted that even though some of the topics covered in the tests were
unknown to the pupils, some of the grade four pupils answered the questions correctly.
Based on an interview with a grade four teacher, the following
are found to be the two major
reasons that grade four pupils could answer the questions that we deemed them to be incapable
of answering. One reason is that the teacher is still implementing
the old curriculum,
which
includes some of the topics that are no longer a part of the present curriculum. The other reason
is that approximately
30% of his pupils attend extra classes and are always ahead of the class.
These factors explain why some pupils in grade four performed better than grade five pupils.
Moreover, it can be said that the learning content is definitely
dependent
on what teachers
decide to teach. The teachers' decisions
might be stronger than the restrictions
imposed by the
course of study, and this would explain some of the exceptions
observed in Tables 15 and 16.
Table 16 provides
some inconsistent
results. The expected order of grades four and five, or of
rural and urban areas, would vary by content domains and by question types.
Data presented in Tables 14, 15, and 16 would be strongly affected by the difficulty
level of
each question item. With reference to data resource, some of the questions that were regarded as
the most difficult ones by all pupils in grades four and five were Q2-2 in "Fraction," Q6-1 1 in
"Measurement," Q6-2 in "Geometry," and Q2-1 and Q2-5b in "Data." Apart from these
questions, Ql-7, Q3-8, Q5-7, Q4-2, Q4-3, and Q6-5 in "Number"; Q2-4 and Q4-9 in "Fraction";
Q3-9 and Q2-9 in "Measurement"; Q5-2 and Q4-7c in "Geometry"; and Q2-5a in "Data" were
regarded as difficult
by grade four pupils,
although grade five students' understanding
of these
questions was somewhat improved. For further analysis of the matter, it is important
to conduct
an analysis on the category of mistakes noted in relation to these questions.
48
International
Cooperation
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Education
Final
(2) Discussion
Report
about the second year survey
The result in Table 19 appears that there is no significant
difference
between the urban and the
rural schools. However, there actually is, when the average is examined by gender. Since the
average score of mail pupils is better than that of female pupils, the uneven numbers of male
and female pupils keep the difference in the scores from sight. Though this tendency appears to
be inconsistent
with the results shown in Tables 12 and 13, we have already found a similar
inconsistency
by content domain in the first year.
In Table 20, the correct answers in some questions are remarkably low in percentage. The.lack
of pupils'
proficiency
in English and their accustomedness to the tests seemed to be the one of
the reasons why the pupils could not answer some questions appropriately.
In order to examine
in what extent English
proficiency
and the accustomedness
to tests affect the result of
mathematics achievement tests, we conducted interview in Newman procedure.
We selected top-five and bottom-five pupils in accordance with the result of continuous
assessment in the class where the achievement test was made in the second year survey. Table
21 shows in which stage the pupils made their mistakes by location. The interview reveals that
the difference
in the type of questions reflects on the steps in which the pupils made mistakes.
In Q6 (1), apart from some difficult terms to read and understand, the number of those who
could explain their strategy to solve the question in English is more than that in other questions.
Some pupils who could not explain their solving process in English
showed their idea in mother
tongue. Even if the pupils could not understand some terms or the setting of the question in
detail, they would be able to guess which operation to use from the term, "add". Though they
could not explain why, many pupils could show their procedural knowledge about addition of
fraction. On the other hand, in Q5 (3) and Q8, it was difficult for the pupils to understand the
meaning of the questions and explain their solutions in English.
Though these questions require
the conceptual
understanding
of fraction, they could only show their guess-work in mother
tongue due to difficulties
in English
expression and.the question settings. Table 22 also shows
that this tendency is more remarkable for the bottom-five pupils than the top-five pupils in their
daily assessment. It suggests that a linguistic
support and a device for posing questions would
be essential
for those pupils
especially
in the questions
which require understanding
of
mathematical concept.
In Table 23, there is almost no difference
in students'
error between the urban and the rural
pupils. It appears to be difficult for pupils to understand the setting of Q5 (3), comparing two
fractions
and answer a symbol which is corresponding
to the correct relation. The tendency is
more obvious for the bottom-five pupils shown in Table 24. In addition,
most pupils explain
their solution in their mother tongue. Though some pupils who tried to draw the diagram for the
comparison of two fractions, their diagram was not accurate. Surprisingly,
a few pupils
answered the fraction with a larger denominator is lager than that with a smaller denominator,
even though their diagram for comparison was correct. The procedure in their solution did not
seem to be logically
structured,
but just a routine work far from the context of the question.
The result in Table 25 shows the high rate of correct answer in Q6 (1). Many pupils seemed to
be able to guess the intention
of the question by the key term of "add", apart from full
understanding
of all the sentences. Table 26 shows that the bottom-five pupils also tended to
answer the correct process of addition.
From Table 27, Q8 appears to be difficult to understand the situation.
Since pupils did not seem
to know the fraction as a ratio, the term of "relationship"
did not link with fraction. As a result,
some pupils who understand the term of "relationship"
tried to show the relationship
of two
quantities
as a difference,
so that their process was subtraction.
Those who understand only the
term of "fraction" arranged a denominator and a numerator with 3 and 5 in an impromptu
manner. Other pupils showed the addition of two fractions. It appeared that they seized on the
49
I nternational
Cooperation
Project
of Mathematics
Education
Final
Report
operation of addition
as an emergent escape root from the difficulty, since it is the simplest
operation to remind. There is no difference
by location,
but by the daily performance. The
top-five pupils tended to show the relationship
at least. Except for only one pupil who could
explain full understanding
of the process in the solution, the difference
between the top-five and
the bottom-five
pupils
seems to depend on reading
competence in English
or the
accustomedness to test. Those factors would affect the level of guess-works.
50
International
2.4
Cooperation
Project
Education
Children
's Final Report
Philippines
MILAGROS IBE
University
of the Philippines
2.4.1
ofMathematics
School
HIRO NINOMIYA
Saitama University
LEVI ELIPANE
Saitama University
Curriculum
The educational ladder in the Philippines
is considered
to
Elementary school education in this country is for a period
can be admitted to secondary school, which lasts for
elementary and secondary school comprise what is referred
2.4.2
Basic
(1)
Schooling
be one of the shortest in the world.
of six years, after which the student
four years. Together, ten years of
to as Basic Education.
Information
Age
A child enters the first year of elementary education at the age of six and is admitted to Grade 1
of elementary school. The child must be at least 5 years and 10 months before the
commencement of the school year. The age of the child is calculated
from his birth date. A
majority of the public elementary schools still do not include preschool,
i.e., below Grade 1.
However, preschool is now offered-usually
by private schools-and
it is for the following
ages and sequence:
Nursery- 3- to 4-yearsold
Kindergarten-
4-
to 5-yearsold
A child who completes Kindergarten and meets the age requirement
Grade 1 in either a public or private school.
(2)
School
Calendar
(6 years)
may go on to
and Examination
The school year comprises 10 months. The Department of Education stipulates
that the school
year must have 200 to 205 school days. The school year usually begins in the first or second
week of June in each calendar year and finishes in the last week of March or the first week of
April of the following
year. The end of the school year is partially
influenced
by the Catholic
liturgical
calendar. Usually, schools end before the Holy Week, although a few schools continue
even after Holy Thursday of the Lenten season. The school calendar permits 2 weeks (12 to 15
days) for break from Christmas till New Year's Day. The summer vacation is for a duration of
two months-April and May.
The year-end examinations are usually conducted from mid-March till close to Easter (this year,
2006, the school year ended on March 24). The National Examination for Grade 6 is conducted
during January or February, if not earlier. The new school year begins on the first Monday in
June in certain schools, otherwise on the second Monday.
T a b l e 1 . S c h o o l c a le n d a r in (y e a r )
J an
F eb
M ar
A p r il
M ay
le
J u ly
Ju n e
A ug
Sep
O ct
N ov
D ec
a
蝣
5 s ?:
51
International
Cooperation
Project
of Mathematics
Education
Children
's Final
Report
(3)
Promotion
System
There is almost 100 percent promotion in elementary school, except in the case of students who
completely
lack the skills required for the next grade. Retention or repeating a grade is more
frequent in the higher grades (Grades 5 and 6) and at the secondary-school
level, not in the first
4 grades in public schools.
In the initial grades, retention of students in the same grade has implications
for denying a place
to incoming students because schools have a shortage of places on account of the large
population.
The students obtain numerical marks for each subject in the grade. The final grade is the
average of the subject grades. In order to pass the grade level, a student must possess an average
grade of 75 in all subjects. The promotion rate is usually 95 percent or higher in Grades 1 to 4
and 85 to 94 percent in Grades 5 and 6. There is seldom 100 percent promotion because of
students who drop out or leave school during the school year.
(4)
There exists a bilingual
policy on the medium of instruction
from Grade 3 upward. Science,
Mathematics, and English must be taught in English;
Social Studies, Filipino
Language, Art,
Practical Arts, Health and Physical Education are taught in the National Language-Filipino.
In
Grades 1 and 2, Health Education is taught in lieu of Science. Formal study of English begins in
Grade 3. However, certain schools-particularly
private ones-begin teaching English in Grade
1; certain schools begin even as early as Kindergarten. Further, the medium of instruction,
particularly
in elite private schools, is English throughout;
however, they have to teach Filipino
(the national
language)
as a subject. Although
there is a bilingual
policy,
it is not fully
observed/implemented.
The reality is that a large number of teachers are not entirely competent
in English
or Filipino;
hence, there is a combination
of languages and switching
of codes
between Filipino
and English.
In Grades 1 and 2, the teacher may use the lingua franca or the commonly spoken local
language, or the mother tongue of the children in the locality.
However, overall, the national
language is becoming widely used, if not entirely, at least side by side with English.
It is
possible that the mother tongue of the teacher is the same as that of the students.
(5)
Class Organization
Grades 1 to 4 are usually class-based;
they are self-contained,
i.e., one teacher teaches all the
subjects-she
is the homeroom teacher. In Grades 5 and 6, certain teachers handle specific
subjects, for example, Math, Science, and English. In the present elementary school curriculum,
the four main subjects-Science,
Mathematics, English,
and Filipino-are
each taught for 60
minutes per day, five days a week. A fifth subject-Makabayan-is
an integration
of Araling
Panlipunan (Social
Studies),
Values, Art, Health and Physical
Education,
and Practical Arts.
Each subject is taught in sessions of 30, 40, or 60 minutes, three or four days a week depending
in the subject. Different grades in elementary schools remain in school anywhere from five to
seven hours a day, depending on the grade level as well as space (rooms and grounds) in the
school.
In the four main subjects, the number of lessons is dependant on the number of class sessions
for the subject per week.
2.4.3
Results
from the Second Year Field Survey
The Philippines
was only able to take part in the Research Team in the second year.
52
International
(1 )
Cooperation
Survey
Project
Development
ofMathematics
Education
Children
's Final
Report
Schedule
Although
the counterpart
researcher from Japan arrived in the Philippines
on December 12,
2005 the field survey could not be conducted immediately
because the schools were closed for
Christmas Break the following
week-school attendance was becoming irregular
due to the
impending
holidays
(Christmas
is the most important
religious
celebration
in Catholic
Philippines).
In the meanwhile, the two researchers conferred and discussed
necessary permissions
were obtained from the two schools.
The actual survey and testing,
January 2006, as follows:
after meeting
with the principals
Table 2. Schedule
D a te
12 /0
13 /0
1 6 /0
1 7 /0
2 3 /0
1/2 0 0 6
1/2 00 6
1/2 0 0 6
1/2 0 0 6
1/2 0 0 6
D
D
D
D
D
ata C
ata C
ata C
ata C
ep a rt
the procedures
of the schools,
for the study;
was scheduled
of data collection
A ctiv ity
o lle ction in th e U rb an S c h o o l (Q u ez o n C ity )
o lle ctio n in th e U rb an S c h o o l (Q u ez o n C ity )
o lle ctio n in th e R u ra l S c h o o l (A p a lit, P a m p an g a)
o lle ctio n in th e R u ra l S ch o o l (A p a lit, P a m p an g a)
u re o f th e resea rch e r to Ja p an
The mathematics classes included
in the survey were conducted for 60 minutes
Monday to Friday and with the same number of lessons for every class meeting.
(2)
Target
Schools
in
per day, from
and Samples
Since over 80% of the enrollment in elementary schools in the Philippines
is in public schools,
the target schools in this study are public elementary schools; private schools are not included.
Further, the schools in the survey included one from an urban area and another from a rural area,
thereby providing
a representation
for both urban and rural schools. Moreover, the rural school
must be one where the mother tongue of the students is not the national language
The schools selected were complete elementary schools, i.e., they have Grades 1 to 6, with over
one section/class
per grade. Grade 4 (or students aged ten or eleven) is the target grade in this
study.
Twoschools-one from Quezon City
province of Pampanga in a municipality
Table
in the National Capital Region and the other from the
50km from Manila-were the testing sites.
3. Location
of schools
S c h o o l lo c a tio n
S a n J o s e , Q u e z o n C ity (in N a tio n a l C a p ita l R e g io n )
A p a lit, P a m p a n g a ( 5 0 k m . fr o m M a n ila )
U rb a n P r im a ry S c h o o l
R u ra l P r im a ry S c h o o l
The urban school is in the capital city, a short distance from the heart of the city. It is an
average-ability
school in an average-and-below-average-income
community. The school also
caters to students from three areas that are occupied by displaced
families.
This school has three
sections for each Grade at the primary level.
The rural school
in an agricultural
is located in a town that is an hour away from Manila,
part of the country.
The two schools
comprise
six grade
levels
(1 to 6). Grades
53
when traveling
1 to 4 are self-contained
in a bus,
classes,
International
Cooperation
Project
Development
of Mathematics
Education
Children
's Final
Report
implying
that the teacher of the class teaches all subjects,
except practical
arts and home
economics; these are taught by another teacher. Since only one class in Grade 4 was selected in
each school, there are only two Grade-4 teachers included in the field survey.
The urban and rural schools have 20 and 16 classrooms,
respectively.
Every classroom has
normal desks, each for two students. However, since the classes have over 50 students each,
certain desks have 3 students seated together.
Table
U r b a n P r im a r y S c h o o l
T o ta l
4. Distribution
G rad e
4
4
H H H H
of sample
G ir ls
28
29
57
B oys
12
12
24
(3)
Results
of Achievement
Test
The Achievement Test comprised ten numbered items. Certain
questions;
therefore, eighteen independent
answers were scored.
The items were either
multiple-choice,
item required
the examinee
Achievement
Test also assessed
to create
short-answer,
a problem
the problem-posing
A ge
9 - 12 y /o
9 - 12 y /o
items
included
T otal
40
41
81
two or three
or supply/constructed
for which
ability
answers. The last
2
the answer was x. Thus, the
of students.
The regular Grade 4 teacher of the class introduced
the researcher (the one from Japan and his
counterpart in the Philippines).
The latter spoke to the students in a mixture of Filipino
and
English
in order to put them at ease. She explained
the testing and assured them that the scores
in the test will not affect their grades in Science or Math.
First, the Filipino
researcher
asked the class to listen to the initial
portion of the English
instruction.
She then asked the children
to translate what they heard in their own words. The
researcher either affirmed or modified the translation.
In the latter part of the test instructions,
the researcher asked the students to read one sentence at a time and translate what he/she read to
Filipino.
This was the researcher's
method for ascertaining
whether or not the children in Grade
4 are able to read the English problems and understand them.
The same procedure
children
in this school
was followed in the rural school. Although
the mother tongue of the
was Pampango, they preferred to translate what they read to Filipino.
The test was assigned after it was ascertained that the anxieties of the students were laid to rest.
There were two or three instances where certain students asked for reassurance with regard to
what they were doing. However, their questions
with regard to answers to the test were not
answered by the researcher.
The testing in the urban school took 40 minutes (10:15
am-10:55
am). The first student to
finish spent only 29 minutes answering the test. There were 40 students in the urban sample (28
girls and 12 boys) and 41 in the rural sample (29 girls and 12 boys).
In both schools, while the students were taking the test, the researcher from Japan interviewed
the teacher in another room; the interview was completed in 20 minutes. In the rural school,
both researchers interviewed
the students.
In the urban school, the students had to be dismissed
at ll:30
am and be back again at 1:30 pm.
The interviews
of the top and bottom five students (as nominated by teachers based on class
achievement),
were scheduled
in the morning of the following
day.
54
International
Cooperation
Project
of Mathematics
Education
Children
's Final
Report
In the rural school, the students remained in school until 2:00 pm. Thus, the interviews were
conducted from 1 to 1:30 pm on the same day. The researchers returned the following
day to
meet with other officials of the school.
A total of 81 students
(Table
were tested-24
boys and 57 girls.
Their
ages ranged from 9 to 12 years
1).
Table 5. Results
A v erag e
of the achievement
B oys
G ir ls
test (%)
H ig h e st
L o w e st
U rb a n ' S c h o o l
5 7 .2 5
5 3 .7 5
5 8 .7 5
9 0
20
4 5 .9 7 5 6 1
4 4 .1 6 6 6 7
4 6 .7 2 4 14
9 5
15
A ll
5 1 .5 4 3 2 1
4 8 .9 5 8 3 3
5 2 .6 3 1 5 8
9 5
15
Table 5 shows that the average score of the samples from the urban and rural schools is 51.5%.
It also shows that students from the urban school scored slightly
better than their counterparts
from the rural school. Further, girls outperformed boys by a small margin with regard to their
performance in the assigned achievement test.
With regard to the range of scores, scores from the rural school
compared with those from the urban school.
achievements
had a slightly
of pupils
larger
in percentage
C o m b in e d
B o y s
G ir ls
C o m b in e d
B oy s
G ir l s
1 0 0 .0
1 0 0 .0
1 0 0 .0
8 7 .5
9 1 .7
8 5 .7
7 8 .0
7 5 .0
8 0 .0
3 7 .3
3 3 .3
3 9 .3
0 .0
0 .0
0 .0
1 0 .0
8 .3
1 0 .7
Q 3 ( l)
1 0 0 .0
1 0 0 .0
1 0 0 .0
9 0 .0
9 1 .7
8 9 .3
6 7 .9
Q i ( i)
(2 )
0 2
range as
Q 3 (2 )
9 0 .2
8 1 .3
9 6 .0
6 5 .0
5 8 .3
Q 4 (l )
9 7 .6
1 0 0 .0
9 6 .0
9 2 .5
1 0 0 .0
8 9 .3
Q 4 (2 )
5 6 .1
5 0 .0
6 0 .0
2 5 .0
2 5 .0
2 5 .0
Q 4 (3 )
7 3 .2
6 2 .5
8 0 .0
5 0 .0
5 0 .0
5 0 .0
Q 5 (l)
6 8 .3
6 2 .5
7 6 .0
6 0 .0
5 8 .3
6 0 .7
Q 5 (2 )
6 8 .3
6 2 .5
7 2 .0
2 7 .5
3 3 .3
2 5 .0
Q 5 (3 )
4 3 .9
5 6 .3
3 6 .0
5 0 .0
5 8 .3
4 6 .4
Q 6 (l)
9 7 .6
1 0 0 .0
9 6 .0
8 0 .0
7 5 .0
8 2 .1
Q 6 (2 )
9 5 .1
8 7 .5
1 0 0 .0
7 2 .5
6 6 .7
7 5 .0
Q 6 (3 )
6 1 .0
3 7 .5
7 6 .0
3 0 .0
1 6 .7
3 5 .7
0 7
6 1 .0
5 0 .0
6 8 .0
9 5 .0
9 1 .7
9 6 .4
Q 8
4 8 .8
4 3 .8
5 2 .0
2 5 .0
2 5 .0
2 5 .0
Q 9
7 5 .6
7 5 .0
7 6 .0
9 5
9 1 .7
9 6 .4
Q lO
7 5 .6
6 2 .5
8 4 .0
5 5 .0
5 0 .0
5 7 .1
The percentage scores for each of the ten questions
indicate
that the items range from very
difficult,
where no correct answer was obtained, to very easy, where the score was 100 percent.
The following
is a brief description
of the performance
of the students
according
to item.
Qi (1) The competence displayed
was close to the level of mastery, particularly
in the urban
school where all the students provided the correct answer. With regard to the rural sample, the
item was easy, but achievement was below 100%.
55
International
Cooperation
Project
Development
of Mathematics
Education
Children
's Final
Report
Qi (2) The item was correctly answered by merely 78% of the students in the urban school and
37.3% of the students from the rural school. Numerous students in the rural school added
instead of subtracting
the two fractions, perhaps thinking
that Ql (2) is of the same nature as Ql
Q2 This item reads "Express the answer in a fraction: 3 å *•E7 =?" None of the students in the
rural school answered this correctly.
Merely 4 students in the urban school
provided
the
3
correct answer-å =•E.
There were at least 20 different variations
of the answer, indicating
that the
students did nof understand the question. It was rather obvious that the students had not yet
related the concept offractions
with that of ratio; they were also unaware that an indicated
or
expressed division
can be expressed as a fraction.
Q3 Answers to this
question
on a bar that is divided
unable to closely
indicated
into four equal
approximate
and can illustrate
3
-7
however, a large number of students
were
that generally
students
portions;
-z on an identical
understand
bar. Although
certain
students
divided
the bar
o
into three portions,
1
these were unequal. What they labeled
as -z was incorrect;
it was more like
1
2°r4-
Q4 In Q4 (1),
almost
all the students
2m bar. However, with regard
in the urban and rural schools
to the next question,
which
were able to shade
asked them to shade
y ofa
1
-m, merely
56% of the students in the urban school and 25% of those in the rural school provided the
correct answer. In each school there was hardly any difference between the answers of the boys
andgirls.
In Q4 (3), as compared with students in the rural school, a higher percentage of students in the
urban school (73.2% versus 50%) correctly shaded VA m of the bar.
Qs In item (1)
the rural
students
school
of this
question,
correctly
68.3%
answered
of the students in the urban school and 60% of those in
4
2
B (i.e., å =•Eis greater than •E=•E.
In item (3), 43.9% of the
in the urban school as compared with 50% of the students
B (that is,
-z is greaterthan
rural samples,
respectively,
j).
In item (2), 68.3%
recognized
that
in the rural schools
and 27.5% of the students
answered
in the urban and
3
1
-z and -z are equal fractions.
Qe In item (1) the word add led almost all the students
2
1
sample to providethe
correct answerto F + F •E
in the urban sample and 80% of the rural
In item (2) under Q6, a high percentage
correctly
In Item (3),
merely
61.0%
of the students
of the urban sample
56
and 30.0%
subtracted
of the rural
2
5
å =from -.
sample
correctly
International
Cooperation
Project
answered the problem
on the total length
Qi This item required
that the decimal
students
in the rural school (95.0%)
students
in the urban school.
of six pieces
of Mathematics
's Final Report
of t m paper in a line.
number 0.7 be changed
correctly
Education
Children
to a fraction.
Almost all the
wrote - ,ascompared with merely 61.0% of the
Qs This item involves expressing the relation between 3kg and 5kg of meat. Merely half the
students in the urban school and one-fourth of those in the rural school answered the problem
correctly.
Q9 The answers from students
of the two schools
one whole; and two wholes out offour
Q10 Three-fourths
(75.6%)
implies
indicated
that they are able to recognize
-z or
-z.
of the urban sample as compared with merely
sample were able to formulate
a sentence problem
55.0% of the rural
2
that leads to the answer -z. A
on fractions
o
majority
of the problems
formulated
involved
adding
two - s or taking away -z from one
o
whole,or
1 from
o
-z.
o
(4)
Results
of Interview
using
Newman Procedure
The top five and the bottom five students in math (based on the performance in the test and
nomination of the teacher) from each of the two schools were individually
interviewed.
Each
interviewee was shown a copy of the test and sequentially
asked questions with regard to Q5 (3),
Q6(1),
andQ8.
The interview was conducted in a mixture of English and Filipino;
however, the students were
told that they can answer/explain
in Filipino.
The responses below are English translations
of
their Filipino
answers.
The interview
questions
roughly
followed
the following
sequence:
Do you remember the test you took this morning/afternoon?
I will show you the test and ask you to read particular
>
>
>
y
>
items.
Can you read Q5 (3) please? (The student reads the item aloud.)
Is there any part (a word or expression)
that you do not understand?
Did you
understand what you read?
Do you remember what you answered in the test? What was your answer?
How did you get that answer? (Please explain how you got the answer.)
Are you sure about your answer? (Do you wantto change it?)
LetusgotoQ6
(l)
Do you understand the question?
Are there any terms you do not understand?
[Ask the interviewee questions in the same sequence]
Proceed to Q8
57
International
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Project
Nowread Q8 aloud.
Do you understand what it is asking
of Mathematics
Education
Children
's Final
Report
for?
The interviewee
explained his answer to each question by showing a solution on a blank sheet
of paper. The interviewer then asked follow-up questions in order to help the student clarify his
answer or solution.
T a b le 7 . T h e p u p ils ' e r ro rs o f p ro b le m
s o lv in g le v e l in u rb a n a n d ru ra l s c h o o ls
N o . o f stu d en ts w ith
C o rre ct A n sw er
F req u en cy o f fa ilu re o n P rob lem so lv in g le v el
I. R ea d in g
II. U n Cd eorstan
n cep dt in g o f
III. P ro c ess
o
z
o
S3
XCB>
DJ-H
Q 5 (3 ) H
0 6 (1)
03
ia-i
Pi
03
蝣
4->
Ho
c
d
JD
DJ-H
U
<3
ht
3
Pi
3
2
10
Q 8
to
-｣4->
4
0
3
7
2
13
F r e q u e n c y o f fa ilu re o n P ro b le m
<L>
u
o
o
l^ ^ cf B ￨^ ^ ^ B = B
蝣
a 2
Jo ,oE
a
L-i
<o
ex
03
L3
e*
5
5
10
a>
o
(S
X
,oU t
f<-DH
Cu
J
c2U-,
<L>
a .
rt<st
・
fi
D
aUi
a
Pi
6
6
1
5
4
5
level according
ll
10
6
N o . o f stu d e n ts w ith
<D
o
I^ ^l B i=B I
(2
<D
sx
CS
4->
｣蝣
to their
C o r re c t A n sw e r
III . P r o c e s s
<D
o
CS
eS
10
ll
15
I I. U n Cd eornsta
c enp dt in g o f
D
o
ca
*e2 ｻ
5
6
5
errors of problem solving
performance
I . R e a d in g
o
｣
c
J3
DJ-1
<L>
o
I^ H B = I
<J3
-4->
e2
u<L>
a .
u<
&
<u
o
I
I
Ui
o
a .
CS
｣
J-t
<D
& ,
Q 5 (3)
0
7
7
0
10
10
10
1
ll
) 6 ( ll
0
2
2
2
9
ll
8
2
10
0 8
5
8
13
6
9
15
4
2
6
From the interviews of students it was evident that reading was not much ofa problem. Students
from Grade 4 were able to read the problems, albeit slowly. A few words were indicated
by
certain students as new or unfamiliar. For example: bracket, relationship,
and ratio. However,
besides these words, everything in the test was judged readable by the students themselves.
It is evident from Tables 8 and 9 that it is in Q8 where a majority of the students from urban
schools faced a difficulty,
even with regard to understanding
the concept itself. This was
because the topic has not been discussed yet in their class. On the other hand, students from the
rural samples also found Q8 challenging
because the topic, according to the students, was
merely discussed in passing-they
have only recently begun studying the particular topic.
Table 8 also indicates that high performers from both the urban and rural schools completely
understood the concept and process required for Q5 (3), and that a majority of them also
understood the concept and process required for Q6 (1).
An observation
of the errors of the students
brings
58
to light
that the students'
understanding
of
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concepts does not imply that they can perform the mathematical
process required to solve a
problem. In the same manner, a student who was unable to correctly work out the mathematical
solution to a problem may be able to provide the correct answer merely by guessing.
Table
9. Pupils'
mistakes
in problem
solving
in Q5 (3)
?5 (3)
U rb an sch o o l
R u ral S ch o o l
L o w p e rfo rm an ce
D iffic u lt w ord s &
E xp ress ion s
"b rack et" an d "A = B "
A ny
rem ark s > T h e
stu d en ts > T h e
stud en ts
> T he
stu d en ts
re ga rd in g p ro b le m
c o n sid ered o n ly
co n sid ere d on ly
co n sid ered o n ly
so lv in g p ro c ess
th e d en o m in ators
th e d en o m in ato rs
th e d en o m in a tors
to id en tify w h ic h
to id en tify w h ich
to id en tify w h ich
fr a ction
is
fra ctio n
is
fr a ction
is
b ig g er.
b ig g er.
b ig g er.
> T h e stu d en ts g ot > T h e stu d en ts g o t
> T h e stu d en ts g ot
c on fu sed
w ith
c on fu se d
w ith
c on fu sed
w ith
th e in struc tion .
th e in stru ctio n .
th e in struc tio n .
> T he
stu d en ts T h e
stu d en ts
T he
stu d en ts
g u esse d .
g u e sse d .
g u esse d .
S p ec ific m istak e s
1
1
1
1
1
1
> 4 = 3
" J - 3
" 4 - 3
*
1
J *
1
3
*
1
4 "
1
3
*
1
4 "
1
3
'5 (3):
The correct answer is 2 or B is greater than A.
The students'
answers were as follows:
"Ais greater because 4 is greater than 3."
"3 is greater than 4; hence, 1/3 is greater than 1/4."
1
1
1
1
"Because - iscloseto
-,and j isbefore
-. Iseeitinaweighingscale."
(A table weighing scale used in the market)
because 3 < 4 when we cross multiply."
"B is greater
3
T
(Other
than
he illustration
interviewees
students
provided
x 1 is lessthan4
illustrations
Answer:
The incorrect
1, they
that 1 of4 parts is smaller
are the
"1" or A is greater than B:
answer was explained
illustrating
x 1, or4 x 1 isgreaterthan3
that depicted
"Since the numerators are both
denominator is the smaller value."
Incorrect
-r y/*^. -.
4
also showed the same cross multiplication,
-r.Theyaddedthat"3
Certain
1\/
1 \/ 14
1
forthe answer is
in the following
59
manner:
that
-z is greater
o
x 1.")
than 1 of3 parts.
same. The one with
the larger
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is greaterthan
-z because4
is greaterthan
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cake is cut into 4, and 77 impliesthatthe
o
1
there are more portions when dividing by 4 so j is greater
considered
the denominators:
since 4 > 3 so j
Similar
explanations
(misconceptions)
were provided
justify the answer "1" (A is greater than B).
Incorrect
Answer "3":
"They're
cake is cut into 3. Thus,
1
is greaterthan
by other
1
-r)the
4
^."
students
in order to
same."
"I provided the answer in haste."
"I thought all the three answers [1, 2, and 3] should
be selected
[he expected
at least once.]
Q 6 (l)
D ifficu lt w ord s &
E xp res s ion s
1
-."
than
A =B
(-r and
o
Table
Report
3."
"t impliesthatthe
4
"Ijust
's Final
that all the 3 options
10. Pupils'
must be selected
mistakes
U rb an sch o o l
in problem
R u ral S ch o o l
2
'tt I"
2
- 1"
> A d d e d b o th th e > A d d ed b oth th e
n u m e rator
an d
nu m erato r an d
d en o m in ator.
d en o m in ato r.
> D id
'c ro ss > C ou ld
not
p ro c ess
d istin g u ish
su b trac tio n '.
b etw e en sim ilar
> C ou ld
not
d istin g u ish
an d
d issim ila r
fr actio n s.
b etw een
s im ila r
an d
d issim ilar > D o e s n ot k n o w
fr ac tion s.
th e co n ce p t at
a ll.
> D o es n ot k n o w
th e c on cep t at all.
S p ec ifi c m ista k e s
2
1
2
1
- + - =
5 + 5 = 10
5
5
10
A n y rem ark s
reg ard in g
p rob lem so lv in g
solving
so I answered
3."
in Q6 (1)
2
V I"
L ow p erfo rm a n ce
a
2
im
d d ed b oth
th e > A d d e d b oth th e
n u m era tor
an d
u m erato r an d th e
d en o m in ato r.
en o m in ato r.
> D id
'cro ss
su b tra ctio n '.
> C o u ld
not
d istin g u ish
b etw een
sim ilar
an d
d iss im ilar
fr a ctio n s .
D o es n ot k n o w
th e c o n cep t at all.
2
1
2
1
- + - =
5 + 5 = 10
5
5
10
1
3
2
5 + 5
4
5 + 5
2
1
4
06 (1):
2
Whenyou add -= Iofwaterto
o
water is in the container?
Those who provided
1
å =Iofwaterin
o
the correct answer stated,
a container, how much
"When the denominators
60
are the same, add only
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the numerators."
Those who provided incorrect answers explained by stating
"Add both numerators and denominators."
Another student stated:
deeper probing,
the following:
2
1
3
= length + - length = ttt length.
showed that the term "liter"
He mistook 1 for length
was not yet introduced
that, on
by the teacher.
One student answered: "Twofive and oneJive of water is three five. " Obviously, he drew from
the concept of adding similar things and was unaware of how to use fractions. However, none of
the students that were interviewed were familiar with the term "similar fractions." They simply
2
1
referred to the fractions
-r- and - as "proper fractions." Obviously
this concept, not the
similarity,
was what was emphasized
One low-performing
by the teacher.
student from the urban school provided
5-2
K^K~5-1
3
T
the following
Answer: 3/4
This is obviously
a wrong carry-over of the early introduction
the values of two fractions.
Certain
students,
after reading
Table
Q (8 )
D if fic u lt w o rd s &
E x p re s s io n s
A n y r e m a rk s
re g a rd in g p r o b le m
s o lv in g p r o c e s s
ll.
mistakes
in problem
solving
for comparing
"1." (liter)
in Q (8)
U rb an sch o o l
R u ral S ch o o l
L o w p e rfo r m a n c e
" b r a c k e t" ,
" k g " , " b r a c k e t" ,
" k g ", " k g " ,
"Jo d y " , "k g " ,
" b o u g h t" ,
" b o u g h t" , " J o d y " , " b o u g h t"
" G eo rg e"
" G e o rg e "
" G e o rg e "
> A d d ed ,
> A d d ed ,
> A d d e d o r d iv id e d > S u b tr a c te d
or
su b tr a c t e d ,
or
s u b tr a c te d ,
or
5 an d 3 .
d iv id e d 5 a n d 3 .
d iv id e d 5 a n d 3 .
d iv id e d 5 a n d 3 . > S p lit 8 in to 5 a n d > G u e s se d
> S p lit 8 in to 5 a n d > T h e to p ic w a s
3.
3.
ju st ta k e n u p in > D o e s
not
> D o es
not
u n d e r s ta n d
th e
p a s s in g .
u n d e rs ta n d
th e > G u e s s e d .
p r o b le m
o r th e
p r o b le m
o r th e
c o n c e p t a t a ll
b e c a u s e it w a sn 't
ta k e n u p y e t .
S p e c ifi c m ista k e s
of the algorithm
the item, stated that they did not understand
Pupils'
solution:
8
co n cep t at
a ll
b e c a u s e it w a sn 't
ta k e n u p y e t.
5 /3 , 1 /3 , 8
8 k g , 5 /3
1 /3 , 2 k g
Q8:
Judy bought 5kg of meat; George bought 3kg of meat. Write afraction in the bracket to
show the relationship
between Judy's meat and George's meat in terms of weight.
"George meat is (
) ofJudy's meat. "
Certain interviewees
stated that they did not understand kg, bracket, and relationship,
provided answers nonetheless. Certain students were unable to read "George" correctly.
61
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"I divided
understand
3 and 5 and got •E=•E";obviously
the student did not quite
o
He used the term "hinati," which means halved. "Hinati ko and 3
why he did this.
3
at 5 kaya -=."If he actually halved 3 + 5 he should
o
express in words his understanding
of "relationship."
have arrived at 4. Perhaps
he was unable to
One student who provided the answer "8" explained, "I just added 5 and 3, 1 don't know what is
asked for." He had answered 3 + 5 = 8.
Another student answered "greater" to indicate that Judy's meat compared with that of George
ismore.
"I divided 5 by 3 or Ijust divided the two numbers to get a fraction."
One student
he subtracted
admitted to not understanding
the problem. His answer was "3 x 2 = 6." Obviously
5 - 3 to obtain 2, and then multiplied
2 and 3.
From the answers to Q8 it is obvious
as a ratio or relationship.
2.4.4
that students
have not yet grasped
the concept
of fractions
Discussion
In the Philippines,
the concept of "fractions" is introduced
as early as Grade 1; this is done
through concrete objects, manipulates,
representations,
and pictures but limited to familiar unit
fractions like
-r,
-z,
and j.4 In Grade 4, some attempt is made at addition
Z.
o
similar
and related fractions;
situations
in this grade.
fractional
parts are introduced
with
and subtraction
concrete
examples
Performance on each of the items was influenced
by one or more of the following
Familiarity
of the item format, illustration,
and directions
for recording the answer(s).
of
and
factors:
For example, with regard to Ql (1) and Ql (2), since both the items are under the heading Ql ,
certain students considered
them both to be involving addition
instead of one on addition and
the other on subtraction.
Mixing multiple-choice
items with supply-type
items is not usually practiced
in schools in the
Philippines;
compounded directions
for one item are not practiced either, as done in Qs(l, 2, and
3). Books in the country would have presented the item in the following
manner:
Study each pair of fractions, answer
"1" if the first fraction is greater
"2" if the second fraction is greater
"3" if the two fractions are equal
Another manner of presenting the item is to direct
between the pair in order to indicate their relationship.
1
1.
2
4
3'
5 - 5
the student
to write the symbol
=, >, or <
4
When dealing with fractions as a ratio, the phrase "write the answer as a fraction" is not usually
used, as done in Q2. Even the textbooks in the Philippines
do not express it in this manner.
Therefore, the low percentage of correct answers to Q2 is largely due to the unfamiliar
manner
of wording the question; the same explanation applies to the low rate of correct responses for Qs.
62
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would have been easier to understand if it was worded as "What fraction
between Judy's meat and George's meat in terms of weight?."
shows
The sequence of Q4 (1, 2, 3) and erroneous reading may have caused the large number of errors
in Q4 (1), Q4 (2), and Q4 (3). It is worth finding out if the percentage of correct answers would
be higher if the sequence hadbeen 1, 3, 2 or3, 2, 1.
Qe (1) was rather easy because of the clue word "add" in the problem.
real understanding,
instead of using "add," which translates
directly
together" can be used.
Perhaps, in order to test
to "+," the phrase "put
Qio is rather challenging;
it actually tests the understanding
of the concept of fractions. Answers
should not be rated on grammar but on evidence ofa sense offractions
and reality math.
Table 3 shows the frequencies and percentages of students who correctly answered each of the
test items in the urban and rural schools, while Table 2 shows the comparative statistics
with
regard to the Achievement Test for the boys and girls as well as the combined group; this is
presented separately
for the urban and rural samples for each item. While the two schools were
selected to represent students from two locations-one from an urban setting and another from a
rural one-the levels of comprehension
are similar for certain items or problems and differ in
other questions. These suggest a difference in the teacher factor rather than a location factor.
The following
are some observations
from the data collected
1. In the urban sample, the students' ability
close to the level of mastery. The students'
as high in the rural sample.
2.
3
-7 ofa length
Ascertaining
an object
4
is divided
in this study:
to add or subtract similar fractions (Ql) is
ability to apply this to word problems is not
is almost mastered;
however, the ability
1
to show - when
o
into 4 equal parts is lacking.
3. In the rural sample, the concepts appeared to be better ingrained. Variations in Q6 were
largely represented by including
and excluding the unit of measure. If judged only in
terms of comprehension of numerical addition or subtraction of similar fractions, it can
be stated that this has been mastered. However, merely 24% of the students included the
unit of measure "1" for "liter" in the answer. The same is true of the answer to the
2
5
problem involving
the subtraction
of -=m from - m.Merely 22 students answered
3
åm
z .The remaining,
4
å ;=
this was obviously
4.
The third
understanding
item
except one, provided
a careless
answered
error.
in Q6 had four main variations
of answers that indicate
a little
1
6
of repeated addition
of six -s. The four variations are lm, 1, - or 1,
0
6
and -z.These four variations
5.
3
the answer å =.The lone exception
together
b
accounted for 61% of the rural sample.
In the urban sample, the answers were more varied. In both the rural and urban samples,
a frequent incorrect
answer was "6 7: ," which most likely
63
is the students'
representation
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f six
-s, or confusion
6
emphasize
the writing
parentheses,
6.
Project
in writing
7 x 6. Perhaps,
the teachers
0
of a multiplication
as in 6 (7: ), to indicate
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do not sufficiently
phrase by using "x," as in (6 x t), or using
a product.
The students have a good understanding
of unit fractions, i.e., fractions with 1 as the
numerator. However, without illustrations
and concrete objects, the students are unable
to visualize the relative sizes of two fractions.
7. In the problems
four students
explained
formulated
by the students
2
-z must be the answer, at least
in which
1
1
2
had a number or word problem, where -r + - = -. This approach was
3
1
2
as correct. Another was j - z - x, which confirmed the
by the students
erroneous concept of addition
and subtraction
of fractions.
8. In Q2, which instructed
the students to "express 3 •E*•E7 as a fraction," the 20
(approximately)
variations of incorrect answers indicated that the students obtain the
answers by operating on the two numbers in a random manner-for example, 7/3 ,
3V7
, 3 x 7, 7 -3-thatprovidedthe
earlier, the students
has not emphasized
2.4.5
Pupils'
answers 2-r, 2 r.l , 21, and 4. As explained
o
are not familiar with this type of item possibly
because the teacher
the concept of a fraction as a ratio or quotient of two numbers.
Errors and Misconceptions
about Fractions
The misconceptions
and causes of erroneous answers were discovered in the interviews,
particularly
with regard to the five low-performing students in each of the two schools.
Misconception 1: To add two similar fractions,
Q^:
1
2
Q.<2):
Misconception
2: To subtract
3
5+5=10
2
Q*^
add their numerators and denominators.
1
3
å
5/+5/=TO7
5
7-7=T4
2
7
two fractions,
subtract
the numerators and denominators.
Q,(2):
5
2_3
7"7"0
Q6(2):
2
5
Cut -mfrom ^m
A
3
nswer: 77
14
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3: The greater the denominator,
greaterthan
the greater
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Hence,
j
is
-r because 4 is greater than 3.
o
Misconception
Hpr*Qi
i c<=»
4:
-r is greater
in
-T
J><r
-r,
4 m ^^">v /*
4x1isgreaterthan3x1.
some impatient
tutor taught the pupils
vvvuujw xxx
Obviously,
a
C
-r > ~j
than
/\3'
-r
the principle:
ifa x d> b x c. However, the students
sequence.
Misconception 5: To add two fractions,
didthe
multiplication
in the incorrect
add all the terms
I5+5"i+5"61=111_7
Certain students
diagonally.
e"g-'
This misconception
Misconception
committed
2
also added
the numbers
1_5+1_6
5+5=2+5=7
appeared to have resulted
6: A unit fraction
e.g., Q6 (3)
the above error. A few students
from Misconception 4 stated above.
added several times is indicated
as a mixed number.
7Tm taken six times is 6-z, which is interpreted
6
6
as 6 x r.
6
Misconception 7: Students do not ascribe much importance to units of measurement like liters
(1) and meter (m). They merely focus on the numbers that they operate on.
Finally, with the use of the Newman method for analyzing the errors committed by the students
on certain items in the achievement test, it can be concluded that the understanding,
or the lack
of understanding
of a student with regard to a concept, does not directly translate to their ability
in working out the mathematical processes. Similarly,
the students' ability with mathematical
processes does not imply that they are able to record their answers correctly or incorrectly.
This
may be explained by the fact that students guess the correct answers, and are affected by the
level of familiarity
with the formulation of the questions.
Further, by analyzing the errors of the students and investigating
the cause for the errors, their
misconceptions
became evident. Being able to perceive the misconceptions
of students with
regard to certain mathematical concepts will enable teachers to understand the manner in which
they should approach and conduct their teaching, while being mindful of these misconceptions.
65
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Thailand
MAITREE
INPRASITHA
MASAMI
Khon Kaen University
2.5.1
of Mathematics
School
1) Historical
1960s
ISODA
University
YUTAKA OHARA
of Tsukuba
Naruto University
Curriculum
List of Major Documents for Grades K-12
National Plan B.E. 2503 (A.D. 1960) enacted in the academic year 1961
1970s
Mathematics
Curriculum
developed
and Technology
1980s
Revised
mathematics
1990s
1 990
Revised
elementary and junior high school curriculum
1990
Revised
senior high school
1999
National
Education
2000s
2001
Curriculum
2002
Amended National
for Promotion of
A.D. 1 978
curriculum
at the senior high school
curriculum
(version
level (A.D. 1981)
(version
A.D. 1978)
A.D. 1981)
Act B.E. 2542 (A.D. 1999)
for Basic Education
of the Mathematics
Standards
(IPST)
by the Institute
Science
2) Features
of Education
Education
B.E. 2544 (A.D. 2001)
Act
Curriculum
for Basic Education
of 2001
for Basic Education
Content Area I: Number & Operations
Standard 1. 1 Understanding
various types of number expressions and using numbers
indailylife
Standard 1.2 Understanding
the effects of number operations
and relations
among
various operations and using operations to solve problems
Standard
Standards
1.3 Using estimation
1.4 Understanding
to compute and solve problems
the number system and using the properties
of number
Content Area II: Measurement
Standard 2. 1 Understanding
basic measurement
Standard
2.2 Measuring
Standard
2.3 Solving
and estimating
things
measure problems
Content Area III: Geometry
Standard
3.1 Explaining
and analyzing
figures
Standard 3.2 Visualization,
spatial reasoning,
Content Area IV: Algebra
Standard 4. 1 Explaining
and analyzing
two- and three-dimensional
geometric
models in problem-solving
various types of patterns,
66
geometrical
relations,
and
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functions
Standard
4.2 Using expressions, equations, graphs, and mathematical models to
represent various situations and interpret their meaning and implement
them
Content Area V: Data Analysis
Standard
5. 1 Understanding
Standard
5.2
Standard
2.5.2
5.3 Using statistics
methods
approach
to analyze data
and probability
and probability
to estimate
for decision-making
phenomena
and problem-solving
& Processes
Standard
6. 1 Ability
to solve problems
Standard
6.2 Ability
to reason
Standard
6.3 Ability
to communicate and represent
Standard
6.4 Ability
to connect mathematical
Standard
6.5 Creativity
Basic
(1)
and using a statistical
Using statistical
rationally
Content Area VI: Skills
& Probability
mathematical
meanings
concepts and other subjects
Information
Schooling
Age
Primary and secondary education in Thailand
comprises six years in a compulsory elementary
program and an additional
six years in secondary school. A child must be enrolled in school by
the age of seven and must attend until age 14.
The educational system is 6-3-3-4. Free public education is compulsory for all children from the
ages of 6-1 5, providing nine years of compulsory education. Pre-school is available
for children
of ages 3-5, primary school for ages 6-ll, lower secondary school for ages 12-14, and upper
secondary for the ages of 15-17. Higher education is generally
provided in a four-year program
for the bachelor's
degree.
(2)
School
Calendar
and Examination
The first semester of the school year begins on May 16 and ends in the first week of October.
After a three-week recess, the second semester begins on November 1, and continues until the
second week of March. The long summer vacation is from the third week of March until May
15, and the cycle is repeated. Classes are held from Monday to Friday, during which time the
students receive approximately
six hours of instruction
each day.
Schools vary in different parts of Thailand.
Small rural schools are not as regimented as the
large city schools nor do they possess as much technological
equipment. Most schools require
uniforms, depending on the affluence of the families.
Students and their families show great
respect for their teachers and appreciate the opportunity
to learn.
T a b le 1 . S c h o o l c a le n d a r in 2 0 0 4
M ay
J une
蝣I!蝣
J u ly
A ug .
Sep.
O c t.
N ov.
蝣
D e c.
Ja n.
Feb.
M a r.
I
67
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There is an end-of-term examination in each school. There is also a special
in Grades 3, 6, and 9 known as the National Test.
Education
Children
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examination
(3)
Promotion
System
The promotion system requires every student to sit for an annual exam in each school and pass
the exam. The passing cut-off is 50% in each subject, without acquiring which the students are
not permitted to proceed to the next grade.
(4)
The medium of instruction
at the primary
children and the teachers are Thai.
(5)
Class
school
level
is Thai.
The mother tongue for the
Organization
Each lesson in primary school is ofa duration of 50 minutes.
Table
2. Class
organization
T h a i L esso n
T im e (M in )
1
T e ac h ers b eg in b y ask in g stu d en ts sh o rt-an sw er q u estion s th at
lea d in to th e d ay 's le sso n or q u e stio n s relate d to e arlier le sso n s;
th e tea ch er m ay also ch e ck h o m ew o rk b y ca llin g stu d e n ts to
sh o w th eir an sw ers in fr o nt o f th e class.
10
T ea ch er d istrib u tes w o rk sh eet w ith sim ilar p ro b lem s. S tu d en ts
w ork in d ep en d en tly .
20
T e ach e r m o n ito rs stu d en ts' w o rk , n otic es so m e c o n fu sio n o n
p articu lar p ro b lem s, a n d d e m o n strates h o w to so lv e th e se
pro b lem s.
T y p ica l fo r te ach er to in terv en e at th e fi rst s ig n o f co n fu sio n o r
stru g g le.
30
T eac h er rev iew s an o th e r w o rk sh ee t an d d e m o n strate s a m e th o d
fo r so lv in g th e m o st ch a llen g in g p ro b lem .
40
T ea ch er c o n d u cts a q u ick oral rev iew o f p ro b le m s lik e th o se
w ork ed o n e arlier.
T e ach er ask s stu d en ts to c o m p lete w ork sh ee ts.
50
U n u su a l to n ot a ssig n h o m ew ork .
A majority of the primary
from ll:30-12:30
pm.
schools
begin
at 8:30
am and continue
till
3:30
pm; lunch
time is
T a b le 3 . S c h o o l h o u r s
T im e
S u b je c t
8 .3 0 9 .3 0
9 .3 0 1 0 .3 0
1 0 .3 0 l l .3 0
l l .3 0 1 2 .3 0
1 2 .3 0 1 3 .3 0
1 3 .3 0 1 4 .3 0
1 4 .3 0 1 5 .3 0
A
B
c
L unc h
D
E
F
2.5.3
Results
from the FirstYear
Field Survey
Thailand has adopted a five-day week and the school semester system. There are two semesters in
a year-the first semester (May-September)
and the second semester (November-March).
Compulsory education is set at six years for elementary school and three forjunior high school.
68
International
(1)
Cooperation
Project
ofMathematics
Education
Children
's Final
Report
Survey Schedule
Table 4. Schedule
D ate
1 1th D ece m b e r 2 0 0 4
12 th D ec em b e r 2 0 0 4
13 th D ece m b er 2 0 0 4
14 th D ece m b er 2 0 0 4
1 5 th D e cem b er 2 0 0 4
16 th D e ce m b er 2 0 0 4
1 7 th D e cem b er 2 0 0 4
1 8 m D e cem b er 2 0 0 4
(2)
Development
of data collection
A ctiv it
A rriv a l at C e nte r fo r R e search in M ath em atic s E d u c atio n (C R M E ),
K h o n K a en U n iv ersity
15 .0 0 - 18 .00 M eetin g on re se arch
A ssist. P ro f. M a itre e In p ra sith a (P h .D .), A ssc.P ro f. Y u tak a O h ara, an d
C R M E C en ter 's Staffs
P rep arin g m a teria ls fo r co llectin g d ata ; T est I & II, V D O d ig ita l
c am era an d so o n .
V isit R u ra l S ch o o l in K h o n K aen an d m eetin g w ith 4 th gra de stu d en ts
0 9 :0 0 V id eo tap in g 4 th g ra d e c lassro o m . (F ra ctio n )
1 0 :0 0 In terv ie w in g th e prin cip al a n d m ath e m atics te a ch er
1 0 :3 0 D istrib u tin g q u e stio n n aire s
1 1 :0 0 T e stin g p art I
12 :0 0 L u n c h
13 :0 0 T estin g p art II
P rep arin g m ate rials fo r co llectin g d ata:T e st I & II, V D O d ig ita l
c am era an d so o n . L eav in g fo r B an g k o k .
V isit U rb an S c h o o l an d m eetin g w ith 4 th g ra d e stu d en ts.
0 9 :3 0 M eetin g w ith th e p rin cip a l an d th e ad m in istra to rs
1 0 :2 0 V id eo tap in g 4 th g ra d e classro o m .(F rac tio n )
1 1 : 1 0 In te rv iew in g th e p rin cip a l an d m ath em atic s tea ch e r
V isit U rb an S c h o o l an d G rad e 4 stu d en ts.
0 9 :0 0 D istrib u tin g q u estio nn aire s
0 9 .3 0 T e stin g P art I
10 :3 0 T estin g P art II
S u m m ary d iscu ss io n at C en ter fo r R ese arc h in M ath em a tics
E d u c ation (C R M E ), F acu lty o f E d u c atio n , K h o n K ae n U n iv ersity .
Target Schools
and Samples
The following criteria were taken into consideration
for the selection of sample schools: school
location (urban, rural) and language used in schools and at home (standard Thai, local language).
As a result, one urban primary school and one rural primary school were selected. In the urban
school, students and teachers always use standard Thai language in classes, while in the rural
primary school rural students and teachers occasionally
use Isaan (local language)
mixed with
Thai language in the classes.
Table
5. Location
of schools
S ch o o l lo c ation
R atch ab u ran a , B an gk o k (C ap ital o f T h ailan d )
N o n g R u a D istrict, K h on K a en (A b o ut 4 4 0 k ilom eters far
fr om B an g k o k )
U rb a n P rim a ry S ch o o l
R u ral P rim ary S ch o o l
Urban Primary School: The selected urban school is situated in Bangkok, the capital of Thailand.
It was founded in 1934 and conducts 24 teaching classes (Grades 1-6). The school is equipped
with five teaching buildings,
including special classrooms for music, arts, agriculture, and
computers and an 8,800-square meter school area. The school is located in a good environmental
area of Bangkok. The numberof teachers and students are 45 and 828, respectively.
Rural Primary School:
The selected
rural school
69
is situated
in Khon Kaen Province in the
International
Cooperation
Project
Development
of Mathematics
Education
Children
's Final
Report
northeastern
part of Thailand.
It is approximately
440 kilometers
away from Bangkok. This
school conducts 15 teaching
classes (Grades 1-6). The school is equipped
with three teaching
buildings,
including
special
classrooms for music, arts, agriculture,
and computers and a
19,200-square
meter school area. The number of teachers and students are 23 and 424,
respectively.
Table
6. Distribution
of sample
G ra d e
4
4
U rb a n P r im a r y S c h o o l
R u ra l P r im a r y S c h o o l
T o ta l
(3)
A ge
(y e a r s )
U rb a n
9
10
ll
A v e ra g e
17
2 1
2
9 .6
Results
of Student
Table
0 -10
books
C o m b in e d
U rb an sch o o l
R u ra l sch o o l
Table
C a lc u la to r
of students'
a le
T o
4
3
7
2
0
2
ta l
0
0
0
age
T o ta l
5
25
22
46
2
9 .7
9 .8
Questionnaire
8. Distribution
3 2 .5 0
1 0 .0 0
9. Distribution
T V
F em
2
2
4
N o . o f P u p ils
R u ra l
l l -2 5
b o ok s
2 2 .8 6
6 8 .5 7
5 2 .5 0
9 0 .0 0
M a le
18
10
28
R a d io
of students'
books (%)
2 6 -10 0
books
7 .1 4
12 .5 0
10 0 b o ok s
1 .4 3
2 .5 0
0 .0 0
0 .0 0
of students'
D esk
Q u ie t
N o
II^
C ro ssed ou t
r e sp o n se
^
S
IH
i ^
H
home items (Yes, %)
D ic t io n a ry
B o ok s
C o m p u te r
( E x c lu d in g
p la c e
te x t b o o k )
C o m b in e d
8 0 .0 0
9 7 .1 4
7 7 .1 4
4 8 .5 7
4 2 .8 6
4 7 .1 4
7 1 .4 2
2 8 .5 7
9 0 .0 0
1 0 0 .0 0
8 7 .5 0
6 5 .0 0
6 2 .5 0
5 7 .5 0
8 5 .0 0
4 2 .5 0
6 6 .6 7
9 3 .3 3
6 3 .3 3
2 6 .6 7
1 6 .6 7
3 3 .3 3
5 3 .3 3
1 0 .0 0
U rb a n
S ch o o l
R ural
S ch o o l
70
International
Cooperation
Project
Table
10. Students'
attitude
of Mathematics
towards
Education
Children
's Final Report
mathematics
A v era g e (
P u p ils ' P e r c e p t io n s
C o m b in e d
U rb a n
R u ra l
1 . 1 u su a lly d o w e ll in m a th e m a tic s
1 .3 4
1 .4 8
0 .9 7
2 . 1 w o u ld lik e to d o m o r e m a th e m a tic s in s c h o o l
1 .2 7
1 .2 0
1 .3 7
3 . M a th e m a tic s is h a r d e r f o r m e th a n fo r m a n y o f m y
2 .7 6
3 .0 8
2 .3 3
c la s s m a te s
4 . 1 e n j o y le a rn in g m a th e m a tic s
1 .3 7
1 .2 8
1 .5 0
5 . 1 a m j u st n o t g o o d a t m a th e m a tic s
3 .2 7
3 .10
3 .5 0
6 . 1 le a rn th in g s q u ic k ly in m a th e m a t ic s
2 .0 1
2 .0 3
2 .0 0
7 . 1 th in k le a rn in g m a th e m a tic s w ill h e lp m e in m y d a ily life
1 .2 7
1 .0 8
1 .5 3
8 . 1 n e e d m a th e m a tic s t o le a rn o th e r s c h o o l su b j e c ts
1 .9 1
1 .7 8
2 .1
9 . 1 n e e d t o d o w e ll in m a th e m a t ic s to g e t in to th e u n iv e rs ity
1 .1 9
1.15
1 .2 3
1 .3 4
1 .2 5
1 .4 7
o f m y c h o ic e
1 0 . 1 n e e d to d o w e ll in m a th e m a tic s to g e t th e j o b I w a n t
æfRemark: 1 =Agree a lot, 2 =Agree a little,
Table
ll. Students'
3 = Disagree
activities
a little,
in their
4 = Disagree
mathematics
a lot
lessons
A v erag e
S tu d e n ts ' a c tiv itie s
C o m b in e d
U rb an
R u ral
2 .1 3
2 .0 5
2 .2 3
1 . 1 p ra c tic e a d d in g , s u b tr a c tin g , m u lt ip ly in g , a n d d iv id in g
2 . 1 w o rk o n fr a c tio n s a n d d e c im a ls
1 .8 6
1 .6 0
2 .2 0
2 .2 6
2 .0 3
2 .5 7
4 . 1 m a k e t a b le s , c h a r ts , o r g r a p h s
2 .1 7
1 .7 8
2 .7 0
5 . 1 le a r n a b o u t s h a p e s s u c h a s c ir c le s , tr ia n g le s , a n d
1 .6 3
1 .4 5
1 .8 7
1 .8 1
1 .6 5
2 .0 3
3 . 1 m e a s u re th in g s in th e c la s sr o o m
a n d a r o u n d th e s c h o o l
r e c ta n g le s
6 . 1 re la te w h a t I a m
le a rn in g in m a th e m a t ic s to m y d a ily
liv e s
7 . 1 w o rk w ith o th e r s tu d e n ts in s m a ll g r o u p s
1 .9 1
1 .8 8
2 .3 0
8 . 1 e x p la in m y a n sw e r s
2 .2 6
2 .18
2 .2 3
9 . 1 lis te n t o th e te a c h e r ta lk
1 .2 3
1.15
1 .3 3
1 .8 1
1 .8 0
1 .8 3
1 1 . 1 r e v ie w
1 .8 4
1 .7 8
2 .9 3
m y h o m e w o rk
1 .5 6
1 .4 5
1 .6 3
3 .4 3
3 .5 3
3 .3 0
*Remark: 1 = Every or almost every lesson,
Table
1.2. Students'
2 = About half the lessons,
attitude
towards
P u p ils ' p e r c e p t io n s
a lot, 2 =Agree a little,
3 = Disagree
71
4 = Never
and teacher
1 .4 6
A v erag e
U rb an
1 .4 0
R u ra l
1 .5 3
1 .3 9
1 .3 3
1 .4 7
1 .2 1
1 .2 3
1 .2 0
1 .1 3
1 .0 5
1 .2 3
C o m b in e d
1 . 1 lik e b e in g in s c h o o l
2 . I th in k th a t stu d e n ts in m y s c h o o l try to d o
th e ir b e s t
3 . I th in k th a t te a c h e rs in m y s c h o o l c a r e a b o u t
th e stu d e n t s
4 .1 th in k th a t te a c h e r s in m y s c h o o l w a n t s tu d e n ts
to d o th e ir b e s t
*Remark: 1 =Agree
school
3 = Some lessons,
a little,
4 = Disagree
a lot
International
Cooperation
Project
Table 1 3. Students'
activities
before
L e s s th a n 1 h o u r
N o
Ite m s
Development
of Mathematics
Education
Children
or after school
1 -2 h o u r s
t im e
's Final
Report
(%)
M o r e th a n 2 b u t
4 o r m o re
le ss th a n 4 h o u r s
h ou rs
1 0 .0 0
4 .2 9
5 8 .5 7
2 0 .0 0
7 .1 4
1 0 .0 0
5 2 .8 6
1 4 .2 9
1 4 .2 9
8 .5 7
l l .4 3
4 1 .4 3
3 0 .0 0
7 .1 4
1 0 .0 0
4 . 1 p la y sp o rt s
l l .4 3
3 5 .7 1
3 0 .0 0
1 2 .8 6
1 0 .0 0
E nj o y m e n t
2 7 .1 4
3 8 .5 7
1 7 .1 4
5 .7 1
1 0 .0 0
5 7 .1 4
1 5 .7 1
1 2 .8 6
7 .1 4
7 .1 4
7 . 1 d o h o m e w o rk
7 .1 4
4 5 .7 1
3 2 .8 6
1 0 .0 0
4 .2 9
1 . 1 w a tc h T V
2 . 1 p la y o r ta lk w ith fr ie n d s
5 . 1 r e a d a b o o k fo r
Table
14. Distribution
of students'
F a m il y m e m b e r
U rb an
R u ral
2 -3 p e r s o n s
1 8 .5 7
2 5 .0 0
1 0 .0 0
4 -6 p e r s o n s
4 4 .2 9
3 5 .0 0
5 6 .6 7
7 -9 p erso n s
2 2 .8 6
2 2 .5 0
2 3 .3 3
7 .14
1 0 .0 0
3 .3 3
M o r e th a n 1 2 p e r s o n s
5 .7 1
2 .5 0
3 .3 3
C ro ss e d o u t
1 .4 3
3 .3 3
of students'
蝣
Table
16. Distribution
C o m b in e d
3 3 .6 2
9
10
right
q u e s tio n s
u m b er
e a su r e m e n t
lg e b ra
ata
a t e rn s , re la tio n s
a n d fu n c t io n s
G e o m e try
T o ta l
19
12
4
4
5
4
48
c o r re c t
9
4
1
4
1
5 3 .7 2
6
4 3 .5 7
5 3 .3 9
25
72
an sw ers
4 9 .2 9
4 2 .8 6
2 4 .2 9
5 8 .9 3
4 7 .1 4
5 4 .5 2
4 6 .1 7
to age
2 0 .5 5
2 7 .5 1
answers by content
S h o r t-a n sw e r
R a te s o f
N o .o f
q u e s tio n s
c o rr e c t
a n sw e r s (% )
5 6 .6 2
5 3 .2 2
4 7 .1 4
6 6 .0 7
L o w e st
3 6 .9 9
16 .44
16 .4 4
answer according
3 7 .4 7
3 6 .8 6
2 9 .4 5
Table 17. Average rates of correct
M u ltip le -c h o ic e
R a te s o f
N o .o f
H ig h e st
8 3 .5 6
6 7 .12
8 3 .56
A v e r a g e r ig h t a n s w e r (% )
U rb a n sc h o o l
3 1 .7 7
2 9 .4 5
ll
answer (%)
6 1.3 9
3 7 .53
5 0 .0 3
of students'
A g e (y e a rs )
right
F em ale
M a le
5 9 .5 9
4 0 .14
5 2 .6 4
A v erag e
6 0 .5 8
3 8 .4 0
5 1.0 8
C o m b in ed
U rb an sch o o l
R u ral sch o o l
N
M
A
D
P
member
C o m b in e d
1 0 - 1 2 p e rs o n s
(4)
family
P u p i ls ' D i s t r i b u t i o n ( % )
domain
T o ta l
N o . of
Q u e st io n s
28
16
5
8
6
10
73
R a te s o f
c o rre c t
an sw ers (
5 2 .9 6
4 8 .0 4
3 5 .7 2
6 2 .5 0
5 0 .4 3
4 9 .0 5
5 2 .8 7
International
Cooperation
Project
of Mathematics
Education
Children
Table 18. Average rates of correct answers of content
U rb an sch o o l
M u ltip le -
Report
domain
R u ra l s c h o o l
S h o r t-
c h o ic e
's Final
T o ta l
M u ltip le -
S h o r t- a n s
c h o ic e
T o ta l
N um ber
6 5 .9 2
a n sw e r
6 2 .2 5
6 4 .0 9
4 4 .2 1
w er
3 2 .0 0
3 8 .l l
5 8 .7 5
5 0 .8 3
5 4 .7 9
4 5 .8 3
3 2 .2 2
3 9 .0 3
A l g e b ra
6 1 .8 8
4 0 .0 0
5 0 .9 4
2 7 .5 0
3 .3 3
1 5 .4 2
D a ta
7 7 .5 0
6 9 .3 8
7 3 .4 4
5 0 .8 3
4 5 .0 0
4 7 .9 2
P a tte rn s ,
6 1 .5 0
6 0 .0 0
6 0 .7 5
4 3 .3 3
3 0 .0 0
3 6 .6 7
5 3 .1 3
6 8 .7 5
6 0 .9 4
3 0 .8 3
3 5 .5 6
3 3 .2 0
re la t io n s a n d
F u n c tio n s
G e o m e try
2.5.4
(1)
Results
Survey
from the SecondYear
Schedule
Table
19. Schedule
D a te
1 4 th N o v e m b e r 2 0 0 5
Target
Schools
Survey
of data collection
A c t iv ity
T e st in g a t th e u rb a n p r im a ry s c h o o l
In te r v ie w in g s tu d e n ts a t th e u r b a n p r im a r y sc h o o l
In te r v ie w in g te a c h e r a t t h e u r b a n p r im a r y s c h o o l
T e st in g a t th e r u ra l p r im a ry s c h o o l
In te r v ie w in g s tu d e n ts a t th e r u ra l p r im a r y sc h o o l
In te r v ie w in g a te a c h e r a t th e ru r a l p r im a ry s c h o o l
2 1 st N o v e m b e r 2 0 0 5
(2)
Field
and Samples
The following criteria were taken into consideration
while selecting the sample schools: school
location (urban, rural) and language used in schools and at home (standard Thai, local language).
As a result, one urban primary school and one rural primary school were selected. In the urban
primary school, students and teachers always use standard Thai language in classes, while in X
the rural primary school rural students and teachers occasionally
use Isaan language (local
language) mixed with Thai language in the classes.
Table
U rb an P rim ary S c h o o l
R u ra l P rim a ry S ch o o l
20. Location
of schools
S ch o o l lo catio n
R atch ab u ran a , B a n gk ok (C ap ita l o f T h a ilan d)
N o n g R u a D istrict, K h o n K ae n (A b ou t 4 4 0 k ilo m eters far
fr o m B an g k o k)
Urban Primary School: The selected urban school is situated in Bangkok, the capital of Thailand.
It was founded in 1934 and conducts 24 teaching classes (Grades 1-6). The school is equipped
with five teaching buildings,
including special classrooms for music, arts, agriculture, and
computers and an 8,800-square meter school area. The school is located in a good environmental
area of Bangkok. The number of teachers and students are 45 and 828, respectively.
Rural Primary School: The selected rural school is situated in Khon Kaen Province in the
northeastern part of Thailand. It is approximately 440 kilometers away from Bangkok. The
school conducts 1 5 teaching classes (Grades 1-6). It is equipped with three teaching buildings,
73
International
Cooperation
Project
of Mathematics
Education
Children
's Final Report
including
special classrooms for music, arts, agriculture,
and computers and a 19,200-square
meter school area. The number of teachers and students are 23 and 424, respectively.
Table
21. Distribution
G ra d e
4
4
R u r a l P rim a r y S c h o o l
T o ta l
Results
(3)
of sample
B oys
13
20
33
G ir ls
17
20
37
of Achievement Test
Table 22. Results of the achievement
A v erag e
G ir ls
T o ta l
30
40
70
A ge
9 .8 7
9 .9
9 .8 9
test (%)
H ig h e s t
L o w est
U rb an S ch o o l
4 3 .2
B oys
4 2 .0 7
4 4 .1 9
75
14
2 6 .1
2 6 .9 5
2 5 .2 5
68
0
A ll
3 3 .4 3
3 3 .1 8
3 3 .6 7
75
0
achievements
of pupils
U ^
H
Q K
H
C o m b in e d
D
8 5 .7 1
9 0 .0 0
(2 )
Q 2
G ir l s
B oy s
in percentage
S c h o o l in r u r a l a r e a
C o m b in e d
B o y s
G i r ls
2 0
2 0
2 0
7 .5
2 5
9 3 .7 5
7 0 .0 0
5 0 .0 0
8 7 .5 0
1 6 .2 5
1 5 .0 0
1 4 .2 9
1 5 .6 3
10
15
5
7 0 .8 3
Q 3 ( l)
9 3 .8 9
9 2 .8 6
9 4 .7 9
6 8 .7 5
6 6 .6 7
Q 3 (2 )
9 3 .3 3
9 2 .8 6
9 3 .7 5
6 1 .2 5
5 6 .6 7
6 5 .8 3
Q 4 (l)
8 3 .3 3
7 8 .5 7
8 7 .5 0
7 4 .3 8
8 3 .7 5
6 5
l l .2 5
1 2 .5 0
10
0
0
0
4 7 .5 0
5 .0 0
Q 4 (2 )
3 .3 3
Q 4 (3 )
6 .2 5
0
1 2 .5 0
9 .1 7
6 .2 5
Q 5 (l)
4 8 .3 3
2 5 .0 0
6 8 .7 5
4 6 .2 5
4 5 .0 0
Q 5 (2 )
1 8 .3 3
2 5 .0 0
1 2 .5 0
1 2 .5 0
2 0 .0 0
0
2 .5 0
0
5 .0 0
2 9 .3 8
3 3 .7 5
Q 5 (3)
7 .14
3 .3 3
0 6 ( 1)
7 7 .9 2
7 4 .ll
8 1 .2 5
3 1 .5 6
0 6 (2 )
3 6 .2 5
3 9 .2 9
3 3 .5 9
1 9 .3 8
2 5 .0 0
1 3 .7 5
1 6 .4 1
1 5 .6 3
2 0 .6 3
1 0 .6 3
0
0
0
0
1 5 .0 0
2 5 .0 0
1 3 .3 3
Q 6 (3 )
9 .8 2
0
Q 7
0
Q 8
2 0 .0 0
3 5 .7 1
6 .2 5
2 0 .0 0
Q 9
4 6 .6 7
5 7 .1 4
3 7 .5
2 1 .2 5
2 5 .0 0
1 7 .5 0
1 0 .7 1
4 0 .6 3
6 .2 5
7 .5 0
5 .0 0
2 6 .6 7
Q l O
(4)
Results of Interview
Table 24. The pupils' errors
using Newman Procedure
of problem solving level in urban
and rural schools
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International
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25 (3)
D iffi cult w ords &
E xpressions
U rban sch ool
"greater than"
"bigger th an"
A ny
rem ark s
regardin g problem
solving pro cess
4
i_ is greater than
4
4
- b ecause 4 is
3
greater th an 3, or
w hen draw ing a
diagram , it also
show s that JL.is
4
_ because 4 is
3
greater than 3 O r
b ecause four
b locks fr om the
diagram is greater
than three b locks
_ b ecause 4 is
3
greater than 3 ,or
w hen draw ing a
diagram , it also
show s that * is
4
R ural School
R ead 2 slash 5 for
tw o over fi ve
"Priebtieb"
(C om pare)
"Te la"(each)
"K aw "(item )
" luak"(select)
is greater than
greater than
Specific m istakes
H igh perform ance
R ead 2 slash 5 for
tw o over fi ve
is greater than
greater than _
3
^
蝣
J^
75
M
3
L ow perform an ce
"Priebtieb"(C om pare)
"cham nu an"(n um ber)
"Tela"(each) "K aw "
(item ) "luak"(select)
"took"(correct)
"suanyai" (m ost)
"term "(fi ll) "y ai kw a"
(b igger than) " 2_ is
5
tw o five"
i.is greater than _
4
3
because 4 is greater
th an 3 .
International
Cooperation
Project
Table 27. Pupils'
Q 6 (D
E x p re ssio n s
U rb an sch o o l
S ay 2 o r 5 fo r
1 , " saab " (k n o w )
5
" k au d " (b o ttle ),
S ay "y ah k -sa i" (n o
m e an in g ) fo r
" y ah k -sab " (to b e
cu rio u s)
mistakes
in problem
R ura l S ch o o l
S ay 1 tu p (sla sh ) 5
fo r 2- o r sa y 2
5
jo o d (p o int) 5 fo r
an d so m e
5
stu de n ts ca n o n ly
re ad so m e w o rd s
ofMathematics
solving
Education
Children
in Q6 (1)
S ay 1 tu p (slash ) 5
fo r
5
蝣
K rab aun k a rn"
(P ro ces s)
e .g ., "T h a" (if)
" L it" (liter)
"N am " (w ater)
"N a i" (in ) " P en "
(is)
A n y re m ark s
re g ard in g p ro b le m
A ll o f stu d en t
k n o w th at if
d en o m in ato rs are
eq u a l, ta k in g
p lu s n u m erato rs
butdo not
un d erstan d
"W h y ?"
T h ere is a stu d en t
try in g to ex p la in
b y d iag ra m b u t
in c orrect an sw er
th e n th ey u se
ad d itio n o f
n u m erato rs
A ll o f stu d e nt
k n o w th at if
d en o m in ato rs a re
e qu a l, tak in g p lu s
n u m era tors b u t d o
n o t u n d erstan d
" W h y ?"
T h ere is a stu d ent
try in g to ex p lain
b y d iag ra m b u t
in c o rre ct an sw er
th e n th ey u se
ad d ition o f
n u m erators
Sp ec ific m istak es
76
's Final Report
M o st o f th e
stud en t's a n sw er
are <L .T hree
5
stu de nt an sw er
3 . stu d ent w h o
10
g et -5-3 th ey d o o n ly
ad d itio n b etw e en
n um erato rs
stud en t w h o g et
J L th ey d o
10
ad d itio n
nu m era to rs an d
nu m era to rs,
L o w p e rform an ce
S ay 2 or 5 fo r
5
" sab " (k n ow ),
"k au d " (b o ttle), S ay
"y ah k -sai" (n o
m ean in g ) fo r
"y a h k -sab " (to b e
cu rio u s), say 2 jo o d
(p o in t) 5 for
2_ .T h ere is a stu d en t
5
w h o ju st o n ly read
th at
"th a" (if) " lit" (liter)
"lo n g " (to g o d o w n )
"y ah k -sab " (to b e
cu rio u s)
" gla i" (to b ec o m e)
an d T h ere is o n e
stu d en t can o n ly re ad
th at "n am " (w ater)
" n ai" (in )
"p en " (is)
O n e o f th e stu d en t
an sw er 1 3 an d
ex p la in b y tak in g
1 + I eq u a 1 13 ; 2
5
5
+ 1 eq u a 1 3 , 5 + 5
e qu al 1 0 th en 10 p lu s
3 eq u al 13 . A n d
a n o th er also an sw er
1 3 b u t can n o t
e x p lain .
International
Cooperation
Project
Table
28. Pupils'
mistakes
in problem
Q (8)
U rb an sch o o l
D iffi cu lt w o rd s
& E x p re ssio n s
l. T h e y
w ron g ly read
so m e w o rd s
e .g .,. sam p u n
(relatio n ) c all to
1 .O n e stu d e nt can
re ad o n ly so m e w o rd s;
'k ilo g ru m " (k ilo g ra m s)
"n a m n ak "(w eig h t)
"p e n "(is)
solving
's Final Report
in Q (8)
H ig h
p erfo rm a n ce
1 . T h e y w ro n g ly read
so m e w o rd s e .g .,
" p aso m p u n " (b reed ) o
"p o ok p u n " (b o u n d )
in stea d o f " sa m p u n "
(re latio n )
2 .O n e stud e n t c an read
p a so m p u n
(b ree d ) an d
pookpun
(b o u n d )
A n y re m ark s
reg ard in g
p rob lem so lv in g
p ro ce ss
S p ecific
m istak e s
Education
Children
o n ly so m e w o rd s e .g .,
"k ilo g ru m " (k ilo gram s)
"n a m n ak " (w eig h t)
"p en "(is)
U rb a n sch o o l
1. T w o stu d en ts
a n sw er
- b ec au se it
5
c an n o t b e
w ritt e n a s 13 _
b eca u se 5
b ig g er th a n 3 .
T h u s, jL is
) -3
n o t a fr a ctio n .
2 . M o st o f th e
R u ra l S ch o o l
1. O n e stu d en t an sw er
_3 b ec au se th e
5
q u e stio n ask to w rite
relatio n sh ip b etw ee n
G eo rg e 's m e at a n d
Ju d y 's m eat th en sh e
w rite G e org e 's m ea t
(3) an d Ju d y 's m eat
(5).
2 . O th er stu d en ts
an sw e r th at L an d 8 :
)
3
th ey ta k e 5 + 3 .
stu d en ts
an sw ere d
th at :!
3
H ig h
L o w p e rfo rm an c e
p erfo rm an ce
1. M o st o f
stu d e n ts d o n ot
un d erstan d
re latio n sh ip
b etw ee n 3 an d
5.
2 . O n e stu d en t
can w rite c o rrect
an sw er a n d
ex p lain fo r
re latio n sh ip
b etw ee n 3 an d
5.
3 . O n e stu d e n t
c an w rite co rre ct
an sw er an d
e xp lain th at th ey
c an n o t w rite
fractio n in term
1. M o st o f th e stu d en ts
d o n o t u n d erstan d th e
relatio n sh ip b etw een 3
an d 5 .
2 . O n e stu d en t ca n
w rite c o rrect an sw er
an d ex p lain th at th e y
c an n o t w rite fr a ction
in te rm s o f 1 .b e cau se
3
5 is b ig g er th an 3 .
3 . T h e y h a v e to
an sw er ｣ .b ec au se 5
3
is b ig g er th a n 3 .
o f _ b ec au se 5
3
is b ig g er th a n 3 .
2.5.5
Discussion
1) Analysis
(1)
of questionnaire
(1st year)
Number of books or possession
of items at home
According to Table 8, the highest percentage of books possessed
was in the range of0 to 10 books (52.50% and 90.00%, respectively).
overall possession of books was in the range of0 to 10 books.
in the urban and rural areas
The highest percentage of
It is evident from Table 9 that people in the urban area have a larger number of items at home
than those in the rural area. In general, people in rural and urban areas possess TVs (Q. V-b,
97. 14%), calculators
(Q. V-a, 80.00%), and radios (Q. V-c, 77. 14%).
77
International
(2)
Cooperation
Students'
蝣
Project
attitude
蝣 ^
ofMathematics
Education
Children
's Final Report
towards Mathematics
蝣
1
1
2
3
4
I w o u ld lik e to d o m o re
m ath em atics in sch o o l
5 4 (7 7 .14 % )
13 ( 18 .5 7 % )
3 (4 .2 9 % )
0 (0 % )
I enjo y lea rn in g m ath em atic s
5 0 (7 1 .4 3% )
14 (2 0 .0 0 % )
6 (8 .5 7 % )
0 (0 % )
5 5 (7 8 .5 7 % )
13 (18 .5 7 % )
0 (0 % )
2 (2 .8 6 % )
57 (8 1 .4 3 % )
ll (15 .7 1% )
0 (0 % )
1 ( 1.4 3 % )
54 (7 7 .14 % )
10 (14 .2 9 % )
4 (5 .7 1% )
2 (2 .8 6 % )
I th in k le arn in g m ath em a tic s
w ill h elp m e in m y d a ily life
I n ee d to d o w ell in
m ath em atic s to g et into th e
u n iv ersity o f m y c h o ice
I n e ed to d o w e ll in
m ath e m atic s to g et th e jo b I
w an t
77.14% of students would like to do more mathematics in school, 71.43% of them enjoy
learning mathematics,
78.57% of them are of the opinion that mathematics will help them in
their daily life, 81.43%
of them believe that mathematics
is important
for them to enter
university,
and 77.14% of them believe that mathematics
is important for them to get the job
they desire. This reveals that the attitude of Thai students toward mathematics is positive.
2) Analysis of test results
The most difficult questions for students were Q. 2-9 and Q. 5-9 and the rate of correct answers
were merely 10%. Fractions, proportions,
and decimal numbers were also difficult
questions for
Thai students. Further analysis will be required on the questions pertaining
to fractions (Q.2-2,
Q.2-4, and Q.4-9) that the Thai students were unable to solve. The rates of correct answers are
as follows.
Q .2 -2
Q .2 -4
Q .4 - 9
1 7 (4 2 .5 0 % )
1 3 ( 3 2 .5 0 % )
1 5 (3 7 .5 0 % )
R u ral S c h o o l
0 (0 % )
1 ( 3 .3 3 % )
4 ( 1 3 .3 3 % )
1 7 (2 4 .2 9 % )
1 4 (2 0 .0 0 % )
1 9 (2 7 .14 % )
A v e ra g e
There are certain differences in students' achievements between the urban and rural areas. This
is caused by various factors. From among these, a few of the major factors are opportunity
for
attending cram school, competitive
environment in urban schools, and access to information.
3) Other
(1)
related
Students'
factors
in response
experiences
in written
to the test
examination
A majority of the students are experienced
when dealing with answering short-answer
submitted a blank paper.
(2)
in answering multiple-choice
tests. Thus,
questions as in this test, a few students
Error in translation
There was an error in Q. 6-5 when translated
into Thai. The meaning of the translated
test was different from its English version. This made it difficult to check the correct
78
International
Cooperation
Project
answer. The test was modified
(3)
Requirement
by the researcher
Education
Children
's Final Report
and reused again.
of further explanation
The explanation provided in the test was not sufficient. For example, certain students
selected the answer code as the number of their family members, although the number
asked for was just before the answer code.
(4)
Timing
of the survey
There were certain topics that students were not taught during the period the data was
collected. As a result, it became difficult for the students to answer the test.
(5)
Language
The language used in classes was different in the urban and rural schools. The students
in the urban school usually use standard Thai language in the class, which is similar to
language used in the test. Local language has an influence
in the interpretation
of
problems or tests. However, students in the rural school, who always use local language
in daily life and occasionally
in classes, faced some difficulty in the interpretation
of the
language used in the test.
(6) Gender
There was no noticeable difference between the results of male and female students.
4) Analysis
of the Results
of the Students'
Achievement
Test (2nd year)
The average score in the achievement test was 33.43%. The performance of the students in the
urban school (average score 43.20%) was better than that of the students in the rural school
(average 26.10%). The performance difference between the two groups was 17.1%
All students from both schools were unable to solve Q7. From the interviews with teachers and
students, it was evident that the students have not been taught the particular
topic as yet. A
majority of the students could not solve Q4 entirely;
however, they succeeded in answering Q4
(1). They solved the problem using the meaning of "a half of the whole. Thus, they were able
to shade _ of2 metres. However, when they applied this idea with Q4 (2), they shaded 1 meter
of the entire 2 meters instead
ofshadingjust
_ of 1 meter.
2
A majority of the students also succeeded in solving addition
and subtraction
of fractions with
equal denominators.
However, all of them did not understand "why" they need not add or
subtract the denominators.
Data from interviews revealed that they have learned as such and
insisted
that it is what the teachers stated. They have a strong belief that they must follow the
advice of the teachers because the teacher is much more experienced than them.
Table 22 presents the overall result of the students' scores in the achievement test. The average
score was 33.43%. The result of the achievement result is not at the expected level. The
performance of the students in the urban school (average score 43.2%) was better than that of
those in the rural school (average score 26.1%). The performance difference
between the two
groups is 17.1%. If the results are examined separately,
it is evident that both boys and girls
from the urban school performed better than those from the rural school. However, there were a
few exceptions. In Q4 (2), rural students obtained 1 1.25%, while urban students obtained only
3.33%; therefore, rural students outperformed urban students. Similarly,
in Q6 (3), rural students
obtained 15.63%, while urban students obtained a slightly
lower score of 13.33%.
The question-wise
analysis of students' achievement reveals that, in general, the performance of
students is comparatively
good in Q4 (1), Ql (2), Q3 (1), and Q6 (1). It appears that to a certain
extent students are familiar with these types of questions as they are similar to those assigned in
79
International
Cooperation
Project
Education
Children
's Final
Report
the classroom test. In general, the most difficult problems were Q6 (3), Q4 (2), Q10, Q5 (2), Q5
(3), and Q5 (l).
It can be assumed that the most difficult questions for students from the urban school were Q7,
Q4 (2), and Q5 (3), and the average scores were 0%, 3.33%, and 3.33%, respectively.
On the
other hand, the most difficult questions for students from the rural school were Q7, Q4 (3), and
Q5 (3), and the average scores were 0%, 0%, and 2.5%, respectively.
According to the teachers'
interviews, the question styles of some of the abovementioned questions were not the same as
the styles of the textbook or classroom tests. Teachers also reported that students are not
accustomed to this type of a questionnaire.
The performance of students from the rural school in Q4 (2) and Q6 (3) was better than that of
students in the urban school. The only reason behind this is that the mathematics teacher of this
class is a network teacher of the teacher who attended the innovative workshop on teaching with
open-approach
method. These two teachers attempted
to provide their students
with the
experience of expressing their ideas freely in class.
On the other hand, the performance of students from the urban school in Q2 is better than that of
students from the rural school. However, the reasons for this are not clear. It is possible
that
students from the rural school lack conceptual understanding
of this topic, which is apparent
from their inability
to relate division
and fractions.
5) Analysis of the Results of Students' Responses to the Interview
We selected ten students for an interview on the basis of their examination
results in
mathematics. Five of them obtained a good score in mathematics and the other five obtained a
low score.
Interview items for the students are divided into the following
four categories:
(a) reading, (b)
understanding
of concept, (c) process, and (d) specific errors (if any).
All students
could were not successful
in comparing
1
and I
4
3
because they had not been
taught this topic as yet. From their attempt, it was evident that they had a rather naive concept
offractions,
as follows. They explained that 1 is greater than 1 because 4 is greater 3.
4
3
Another explanation
was provided with the diagram of two circles that were divided into four
and three equal parts-since
four parts are more than three parts, consequently,
1 is greater
thanI.
3
Data from the interview revealed that a majority of the students were able to solve problems
similar to those they have learned earlier. However, when they are questioned "why did you do
that?", their responses were something like this: "The teacher told us to do so." The students do
not really understand "why" the strategy works. For example, when adding fractions with equal
denominators, they recite that the result is obtained by adding the numerators and taking the
denominators. They are unable to reason why only the numerators are added but not the
denominators.
Table 24 present the levels of errors in problem-solving
in urban and rural schools. According to
the findings, all students were able to read Q5 (3), Q6 (1), and Q8, but none of them could
understand the concept ofQ5 (3) and Q8, and only a few were able to understand the concept of
Q6(l).
With regard to process skill, in Q5 (3), a majority of them were unable to show the correct
process. In Q6 (1) and Q8, half of the students were unable to show the correct process. In
80
International
Cooperation
Project
comparison, a greater number of students
than those from the rural school.
of Mathematics
from the urban school
Education
Children
's Final Report
were able to solve problems
Table 25 shows the levels of problem-solving
errors according to their performance. According
to the findings, all students were able to read Q5 (3), Q6 (1), and Q8; none of them were able to
understand the concept of Q5 (3) and Q8, and only a few of them were able to understand the
concept ofQ6 (1).
High-performers
performed slightly
better than the low performers, except in Q8 where the
latter group committed a large number of errors. In comparison, the high performers were able
to solve problems better than the low performers, except in Q8.
In Q5 (3), the students from both schools found it difficult
to read the words "correct" and
"appropriate."
They committed a few errors in the process skill. For example, according to them,
JL is greater than J_ since 4 is greater than 3, and some stated that ifa denominator is greater
4
3
than other denominators
then that fraction
is greater than other fractions,
etc.
In Q6 (1), the students of the rural school found it difficult
to read the word "process." They
committed a few errors in the process skill. For example, they added both the numerators and
denominators in order to obtain the solution.
In Q8, the students of both schools found it difficult
to read the words "relationship,"
"in terms
of," and "bracket." They committed a few errors in the process skill. For example, they only
wrote the answer-L -without showing the process.
To conclude, it appears that lack of sufficient orientation
to various types of problems, the
so-called
only textbook-dependent
lesson plan, the tendency of copying problems from
textbooks for a test, and lack of adequate stimulation
for innovative thinking
are the main
obstructions
for students in achieving a satisfactory
level in mathematics learning.
References
Department of Curriculum and Instruction
Development, Ministry
of Education,
Basic
Education
Curriculum
B.E. 2544
(A.D 2001).
Bangkok
: The
Transportation
Organization
of Thailand,
2002.
Inprasitha,
M. ( 1997). Problem solving: a basis to reform mathematics
the National Research Council of Thailand, 29 (2), 22 1 -259.
instruction.
Thailand.
Express
Journal
of
Iwasaki, H. (2005).
Empirical Study on the Evaluation Methodfor International
Cooperation
in Mathematics Education in Developing
Countries:
Focusing on Pupils' Learning
Achievement. Hiroshima:
Graduate school of International
development
and cooperation,
Hiroshima University.
The Institute
for the Promotion of Teaching Science and Technology, Ministry of Education.
(2003 ). Mathematics Textbook for 4th grade. Kurusapa Ladprao Publishing,
Bangkok.
Thai Educational System. Retrieved January 1 0, 2005,
http ://www. magma.ca/~thaiott/edsystem.
htm
81
from
International
2.6
Cooperation
Project
Education
Children
's Final Report
Zambia
Toyomi Uchida
Satoshi
Hiroshima
Hiroshima
2.6.1
of Mathematics
University
School
Nakamura
Bentry Nkahta
University
University
of Zambia
Curriculum
The latest mathematics syllabus for Grades 1-7 in Zambia was published
in 2003 and that for
Grades 10-12 in 2002, although that for Grades 8 and 9 has not been renewed since 1983. The
general aims of the syllabus for Grade 1-7 are as follows.
1.
To equip the child with necessary knowledge and skills to enable him/her to live effectively
in this modern age of Science and Technology
and to contribute
to the social and
economical development of Zambia.
2. To stimulate
and encourage creativity
and problem solving.
3.
To develop the mathematics abilities
of a child to his/her full potential
and assist
study Mathematics as a discipline
and to use it as a tool in various subject areas.
him/her
4.
To assist the child
comprehend his/her
may better
5.
To develop
in the child
6.
To develop
interest
7.
To build
up understanding
and appreciation
of basic
computational
skills in order to apply them in everyday life.
8.
To develop clear mathematical
thinking
9.
To develop
problems
10. To develop
ability
to understand
environment.
mathematical
an appreciation
in Mathematics
to recognize
13. To provide
in order that he/she
of Mathematics
in the traditional
and encourage a spirit
of inquiry.
and expression
environment.
mathematical
concepts
and
in the child
and to solve them with related
and foster order, speed and accuracy.
1 1. To provide the child with the necessary
him/her to be productive and self reliant.
12. To develop
self-reliance.
concept
to
in the child
the necessary
a positive
mathematical
attitude
mathematical
towards
pre-requisites
(Mathematics
knowledge
production,
for further
Syllabus
and skills
in order
entrepreneurialship
for
and
education.
Grade 1-7, 2003,
pp.5,6
The mathematics syllabus in Zambia has the feature of being like a "spiral"; the same topics can
be found in the syllabus of several grades and taught repeatedly in the period of9 years of basic
education. This appears to be the effect of the British Mathematics Curriculum that was adapted
during the "Modernization
of Mathematics Education."
2.6.2
Basic
Information
(1)
Schooling
Age
According to Zambian policy, children
in practice, the ages at which children
are expected to begin schooling at the age of 7. However,
enter school vary because in certain cases, children who
82
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Project
ofMathematics
Education
Children
's Final Report
reach the age of 7 are unable to find seats in schools, particularly
in urban areas; while in certain
cases parents are unable to raise money for school requirements
and hence delay sending the
child to school. In rural areas, certain children live at a substantial
distance from a school and
their parents only send their children when they are old enough to walk long distances to school.
As a result of this, there is a wide range in terms of age of children within a class.
(2)
School
Calendar
and Examination
The school year in Zambia begins in January and ends in December. The school year is divided
into 3 terms, as shown in table 1, and each term has a duration of 3 months. There is a
one-month break between each term.
T a b le 1 . S c h o o l c a le n d a r (y e a r)
Ja n .
F eb .
M a r.
A p ril
M ay
1 st T e rm
Jun e
Ju ly
A ug .
S ep .
2 nd T e rm
O c t.
N ov .
D ec.
3 rd T e rm
At the end of Grade 7, students take a national examination and those who are able to attain the
cut-offmarks proceed to Grade 8. All items in the Grade 7 examination are multiple-choice and
presented in English. Similarly, at the end of Grade 9, students take a national examination and
those who qualify proceed to Grade 10. The plans are that in future, if there are sufficient
numberof seats available in Grade 8, students fromGrade 7 must be promoted automatically to
Grade8.
(3)
Promotion System
The school system in Zambia is changing fromthe system of 7 years of primary education, 2
years of junior secondary education, 3 years of senior secondary education, and 2-7 years of
tertiary (higher) education to 9 years of basic education, 3 years of high school education, and
2-7 years of tertiary (higher) education.
This change is still underway; therefore, two school systems are coexisting at present. The speed
of this school reformation is rather slow in rural areas; as a result of this, there are schools in
these areas that continue to have only lower-basic (Grades 1-4) and middle-basic (Grades 5-7)
grades, although they are knownas basic schools.
Table 2. Education
a
m
m
蝣m
蝣
H i;ｫｰm
m
1
m H I
3 y e ars
L o w e r)
(4)
tｻ
m M88
I m m 蝣
P rim ar y S c h o o l
7 y e a rs
I
system in Zambia
u c a tio nu s
1
m m m u c atio n I
B a s ic S c h o o l
4 y e a rs
(M id d le )
蝣I
蝣H
蝣
S e c o n d a ry S c h o o l
2 y e a rs
3 y e a rs
(L o w e r)
(U p p e r)
1I
蝣 1 I ￨>
rm
1i
H ig h S ch o o l
2 y e a rs
3 y e ars
(U p p er)
The officiallanguage in Zambia is English, although there are approximately 72 tribes that have
their ownlanguage (dialect), which is usually spoken in their homes. English is also the medium
of instruction;
therefore, teachers basically
conduct lessons in every subject in English.
However,students, particularly in the initial years of schooling, usually speak local languages
outside classes and school. Thus, students-particularly
in lower grades-face difficulty in
83
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understanding
explain certain
understand the
In certain cases,
2.6.3
Results
Project
or speaking English.
Occasionally,
certain
things to students;
however, the problem
local language and therefore will not be able
the teacher may not know how to speak the
from the FirstYear
Field
of Mathematics
Education
Children
's Final
Report
teachers use a local language to
is that certain students may not
to communicate with the teachers.
main local language.
Survey
(1 )
Survey
Schedule
A one-mandelegation
visited Zambia from January 17, 2005 to January 27, 2005 in order to
facilitate
the field survey with his Zambian counterpart. This implies that data collection
was
conducted just two weeks after the commencement of the first term. The detailed schedule for
data collection
is tabulated as follows.
Table
D a te
16 th / J an / 2 0 0 5
17 th- 1 9 th / J a n / 2 0 0 5
2 0 th - 2 7 th / J a n / 2 0 0 5
2 8 th / J an / 2 0 0 5
(2)
Target
Schools
3. Schedule
of data
collection
A c tiv ity
A rriv a l in L u sa k a , Z a m b ia
P r ep ara tio n o f th e su rv ey
D isc u ssio n w ith D istric t E d u c a tio n O f fi c e s a n d
T a rg ete d S ch o o ls
D ata c o llec tio n in tw o s am p le sc h o o ls in L u sa k a
P ro v in c e a n d th e ir su b se q u en t rem e d ia l w o rk
D e p artu re fr o m L u sa k a , Z a m b ia
and Samples
[1 ] Sampling procedures
From amongthe 9 provinces in Zambia, Lusaka-the capital province-was selected as a
sample due to the time constraints for completing data collection. According to the Central
Statistic Office, the extra-departmental organization under the Finance Ministry, an urban area is
defined by 3 criteria-population
size, economic activity, and facilities
available in the area.
Taking into consideration these criteria, from among4 districts in the province, Lusaka District
wasselected to represent the urban area and Chongwe District to represent the rural area.
For the selection of average schools in both urban and rural areas, the survey team requested
each District Education Office (DEO) to recommendappropriate schools. From the discussion
with District Education Standards Officers (DESOs), we selected one average government
school each fromboth urban and rural districts. The criteria taken into consideration were as
follows:
a. Ranking of the sample schools according to the results of National Final Exam,
b. Social and economical strata in the catchment area of the sample schools.
However, the survey team was unable to go through the actual data related to the
abovementioned criteria.
For the selection of a sample Grade 4 class in the urban average school, it was reported that
classes are not formed according to the results of the students. Thus, they are all assumed as
being uniform and one sample Grade 4 class was selected since the timing of its lesson was able
1 In addition to this definition, an urban area must have a minimumpopulation size of5,000 people. The main
economic activity of the population must be non-agricultural, such as wage employment. In addition, the area must
have basic modern facilities such as piped water, tarred roads, post office, police station, health facility, etc.
84
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Development
of Mathematics
Education
Children
to fit into the overall schedule for data collection.
In the average rural school,
Grade 4 class; thus, all the Grade 4 students have been targeted in this survey.
[2] Basic
school
's Final
Report
there is only one
information
As mentioned earlier, Zambia is currently transforming
its school system from a 7-year primary
and 5-year secondary education to a 9-year basic and 3-year high school education. Particularly
in remote areas, there remain a few basic schools that accommodate only 4 grades due to a
number of constraints.
Due to these situations,
there are various sizes of basic schools
depending on the level of progress in their transformation;
two of our sample schools are not
exceptional.
The following
table
presents
the details
Table
4. Location
of schools
S ch o o l lo ca tio n
2 k m fro m th e c a p ita l c ity , L u sa k a
5 8 k m fro m th e c ap ita l c ity , L u sa k a
1 3 k m fro m D istric t c e n tre
[3] Size of samples
of the sample schools.
and their distribution
by school location,
sex, and age
As mentioned earlier in sample procedure, we selected one class each from both urban and rural
average primary schools. In the rural average school, there was only one Grade 4 class;
therefore, we decided to include all the Grade 4 students in the class.
There were certain situations that made it difficult to determine the number of sample students
in each school. Prominent amongthese situations were
a. Timing: The timing of our data collection overlapped with the registration period for
both schools. This resulted in variability
in the number of sample students.
There werea few changes in the number of students due to the influx and
out flux of repeaters from Grade 5 and to Grade 3, respectively.
b. Weather: The month of January is a part of the rainy season in Zambia. As a result,
students' attendance actually depended on the weather. In the rural school,
a number of students remained absent until the second day of our data
collection due to the heavy rain in the morning.
Nonetheless, the following tables present the number and distribution
of sample students by
location, sex, and age. Due to the Government's policy of age restriction for school admission
and existence of repeaters and dropouts, age distribution
appears very wide and the peak is
toward the older age group from the expected enrolling age of 10.
of sample
G rad e
4
M a le
29
4
16
45
T o ta l
85
F e m a le
T o ta l
21
17
38
50
33
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Development
A ge
(y e a rs)
8
9
10
ll
12
13
14
15
A v e ra g e
(3)
Results
of Student
U rb a n
0
1
3
g
7
20
9
2
12 .5 y rs
of Mathematics
of students'
N o . o f P u p ils
R u ra l
0
1
3
14
2
10
1
2
l l .8 y rs
Education
Children
's Final
Report
age
T o ta l
0
2
6
22
9
30
10
4
12 .3 y rs
Questionnaire
Questionnaires
were administered
to the students by explaining
the English version of each item
in the questionnaire
in Nyanja, which is the local language of instruction
in the targeted districts.
This method was agreed upon after observing that a majority of the targeted students did not
possess
sufficient
reading
comprehension
in English.
After having
administered
the
questionnaire,
the survey team also conducted a little remedial work individually
with certain
students in order to fill out the unanswered items or to confirm incorrectly
marked items. In the
rural school, the survey was conducted through individual
interviews
of the targeted students,
and questionnaires
were filled out by the Zambian counterparts. The followings
are the major
findings from the students' questionnaires.
[1]
Frequency
of use of English
and Nyanja (local
language
of instruction
in the area)
Unlike other participating
countries, the survey team decided to question the frequency with
which students use both the official and local languages of instruction
in the targeted districts.
In
Zambia, in addition
to the official language-English-7
major local languages
have been
selected as local languages of instruction
in schools. Considering
the composition
of tribes in
the concerned areas, one of these local languages was carefully decided upon and used in the
schools. This arrangement can be said to be a sort of compromise as there are 73 tribes in the
country. Therefore, it is possible
that there are certain students whose mother tongue is different
from the local language used in the schools and this situation
may affect the achievement of
students. In order to examine this influence, we had added an additional
question to the
questionnaire.
As shown in Table 5, only approximately
5% of all the targeted
students answered that they
either always or almost always speak English at home; all these students are from the rural
school. This result may require some verification
as it does not coincide with the achievement in
mathematics tests.
On the other hand, a difference
in the percentage of students who occasionally
speak English
can be explained by the difference in the number of opportunities
these students have to interact
with people other than their family members. It can be supposed that students in urban areas
have a greater opportunity
to interact with people around their house, in the market, etc.
With regard to the reasons behind the difference
in the frequency in use of Nyanja between
urban and rural schools, one reason is the uneven distribution
of either fathers' or mothers'
tribes in each school. 70-80 % of the students in the urban school have parents who are either
Nyanja or its tribal cousin Benba; this figure is only approximately
30% in the rural school.
86
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of Mathematics
Education
Children
's Final
Report
Similar to the case of speaking English, a difference in the level of interaction
with people
outside their houses can also contribute toward the difference in frequency of the use ofNyanja,
as it is the most popular language of communication in public places in these districts.
Table
7. Pupil's
distribution
on frequency
A lw a y s
in use of English
A lm o st
by location
S o m e t im e s
N ev e r
(%)
A lw a y s
C o m b in e d
1 .2
3 .7
6 5 .9
2 9 .3
U rb a n
0 .0
0 .0
7 6 .0
2 4 .0
R u ra l
3 .1
9 .4
5 0 .0
3 7 .5
Table 8. Pupil's
distribution
on frequency
A lw a y s
in use of Nyanja
by location
S o m e t im e s
N ever
A lm o s t
(%)
A lw a y s
C o m b in e d
4 2 .7
2 5 .6
3 0 .5
1 .2
U rb a n
4 4 .0
3 4 .0
2 0 .0
2 .0
R u ra l
4 0 .6
1 2 .5
4 6 .9
0 .0
Table
9. Pupil's
distribution
on their
fathers'
tribe
by location
(%)
N y a n ja
B em b a
T o ng a
L o zi
S o li
N d e b e le
L e nj e
O th e rs
C o m b in e d
4 1 .0
2 0 .5
2 .4
7 .2
4 .8
1 2 .0
4 .8
7 .2
U rb a n
5 0 .0
3 2 .0
4 .0
1 2 .0
0 .0
0 .0
0 .0
2 .0
R u ra l
2 7 .3
3 .0
0 .0
0 .0
1 2 .1
3 0 .3
1 2 .1
1 5 .2
Table
10. Pupil's
distribution
on their
mothers'
tribe
by location
(%)
N a a nj a
B em b a
T on ga
L o zi
S o li
N d e b e le
L e n ie
O th e rs
C o m b in e d
4 6 .3
1 3 .4
l l .0
4 .9
1 .2
7 .3
4 .9
l l .0
U rb a n
5 8 .0
1 6 .0
1 6 .0
6 .0
0 .0
0 .0
2 .0
2 .0
R u ra l
2 8 .1
9 .4
3 .1
3 .1
3 .1
1 8 .8
9 .4
2 5 .0
[2] Possession
of books and domestic
items at home
Over 70 % of the targeted students in both urban and rural schools have 10 or less than 10 books
at home. A majority of the targeted students in both schools have radios at home, while almost
none of them have computers.
Table
ll.
Distribution
of students'
books
0-10
bo ok s
l l-2 5
2 6 -1 0 0
B o ok s
b ook s
100 bo ok s
C o m b in e d
7 3 .5
1 9 .3
4 .8
2 .4
U rb a n sc h o o l
7 2 .0
1 6 .0
8 .0
4 .0
R u ra l sc h o o l
7 5 .8
2 4 .2
0 .0
0 .0
87
(%)
N o
re sp o n se
M
0 .0
IH
0 .0
0 .0
C ro s s e d
I^
H
o ut
H
^
M
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Table
12. Distribution
C a lc u la t o r
C o m b in e d
U r b a n sc h o o l
R u ra l sc h o o l
T V
Development
of students'
R a d io
D e sk
ofMathematics
Education
Children
's Final Report
home items (Yes, %)
B o ok s
Q u ie t
D ic t io n a r y
( E x c lu d in g
C o m p u ter
2 .4
p la c e
3 9 .8
5 6 .6
9 1 .6
3 9 .8
5 0 .6
3 4 .9
te x t b o o k )
5 6 .6
4 8 .0
7 2 .0
8 6 .0
3 8 .0
6 8 .0
5 2 .0
5 2 .0
4 .0
9 .1
6 3 .6
0 .0
2 7 .3
Table
3 3 .3
1 0 0 .0
4 2 .4
1 3. Students'
attitude
2 4 .2
towards
mathematics
A v e ra g e
P u p ils ' P e rce p tio n s
C o m b in e d
U rb a n
R u ra l
1 . 1 u su a lly d o w e ll in m a th e m a tic s
2 .0
2 .0
2 .1
2 . 1 w o u ld lik e to d o m o re m a th e m a tic s in sch o o l
1 .5
1 .4
1 .7
2 .2
2 .1
2 .4
3 . M a th e m a tic s is h a rd e r fo r m e th a n fo r m an y o f
m y c la ssm a te s
4 . 1 en jo y le a rn in g m ath e m atic s
1 .4
1 .4
1 .5
5 . 1 a m ju st n o t g o o d at m ath em atic s
2 .5
2 .1
3 .2
6 . 1 le a rn th in g s q u ic k ly in m a th e m a tic s
1.8
1 .6
2 .0
1 .9
1 .6
2 .4
1 .8
1 .4
2 .5
1 .6
1 .4
2 .0
1 .7
1 .3
2 .0
7 . I th in k le a rn in g m a th e m a tic s w ill h e lp m e in
m y d a ily life
8 . I n e ed m ath e m atic s to
le arn
o th e r sc h o o l
su bj e c ts
9 . 1 n ee d to d o w e ll in m ath em atic s to g et in to th e
u n iv e rs ity o f m y c h o ic e
1 0 . I n e e d to d o w e ll in m a th e m a tic s to g et th e
jo b I w a n t
The results presented in the above table reveal that students tended to have positive attitudes
toward learning
mathematics
in schools as statements such as "I would like to do more
mathematics in school," or "I enjoy learning mathematics" are valued positively.
If we compare
the perceptions
of urban students with rural students, it may be stated that students in the rural
school tended to be more neutral toward learning of mathematics than those in the urban school,
particularly
on questions regarding the meaning of learning mathematics in or outside school.
There is also an inconsistency
in the urban students'
perceptions
toward their ability
in
mathematics. The positive statement of "I usually do well in mathematics,"
and negative ones
such as "Mathematics is harder for me than for many of my classmates," and "I am just not
good at mathematics" are equally perceived by students in the urban school.
[3] Frequency
of students'
activities
in their
mathematics
lessons
The targeted students were asked to reveal the frequency of the following
activities
in their
mathematics
lessons according to f4 scales-1:
every or almost every lesson, 2: about half of
the lessons, 3: some lessons, or 4: never.
The table below shows that regardless
of the location of the schools, a majority of the activities
were not frequently conducted in class, except the following
3: "I practice adding, subtracting,
multiplying,
and dividing
without using a calculator," I listen to the teacher talk," and "I work
problems on my own."
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Table 14. Students'
activities
in their
of Mathematics
mathematics
Education
Children
's Final
Report
lessons
S t u d e n ts ' a c t iv itie s
A v e ra g e
C o m b in e d
U rb an
R u ra l
2 .1
1 .3
3 .3
3 .0
2 .7
3 .3
2 .9
2 .8
3 .2
3 .1
2 .8
3 .5
2 .2
1 .7
2 .9
2 .5
2 .0
3 .4
7 . 1 w o r k w ith o th e r s tu d e n t s in s m a ll g r o u p s
2 .6
2 .0
3 .5
9 . 1 lis te n to th e t e a c h e r ta lk
2 .5
2 .0
3 .3
1 .3
1 .4
1 .1
1 0 . 1 w o r k p r o b le m s o n m y o w n
1 .8
1 .7
1 .9
1 1 . 1 r e v ie w m y h o m e w o r k
2 .2
1 .8
2 .9
3 .0
2 .7
3 .5
1 3 . 1 u s e a c a lc u la t o r
3 .4
3 .0
4 .0
1 . I p r a c tic e
a d d in g , s u b tr a c t in g , m u lt ip ly in g , a n d
d iv id in g w it h o u t u s in g a c a lc u la to r
2 . 1 w o r k o n fr a c tio n s a n d d e c im a ls
3 . I m e a s u r e t h in g s in th e c la s sr o o m
sch o o l
a n d aro u n d th e
4 . 1 m a k e ta b le s , c h a r ts , o r g r a p h s
5 . I le a r n a b o u t s h a p e s s u c h a s c irc le s , tr ia n g le s , a n d
r e c t a n g le s
6 . I r e la t e w h a t I a m
le a rn in g in m a th e m a tic s t o m y
d a ily liv e s
[4] Students'
perception
of the school
and teacher
Students were asked to rate their level of agreement with the following perceptions
four scales-1:
agree a lot, 2: agree a little, 3: disagree a little, or 4: disagree a lot.
according
to
The table below indicates
that students in both locations
have a positive opinion of their
teachers and schools. On the other hand, students themselves believe that they are making an
effort to do better in spite of their teachers' efforts and care.
Table
15. Pupils'
perception
towards
school
P u p ils ' p e rc e p tio n s
and teacher
A v e ra g e
C o m b in e d
U rb an
R u ra l
1 . 1 lik e b e in g in sc h o o l
1 .0
1 .0
1 .1
2 . I th in k th at stu d e n ts in m y sch o o l try
to d o th e ir b e st
1 .9
2 .0
1 .8
1 .1
1 .1
1 .1
1 .1
1 .2
1 .1
3 . 1 th in k th at te a ch e rs in m y s ch o o l c a re
a b o u t th e stu d e n ts
4 .1 th in k th a t tea c h e rs in m y sc h o o l w a n t
stu d e n ts to d o th e ir b e st
[5] Time spent on other activities
before or after school on a normal school day
According to the results shown below, approximately
40% of the students have no time to watch
TV before or after school on a normal day. Almost 90% of the students do not have time to use
the computer, and these figures are clearly related to the level of possession
of these items.
On the other hand, over 40% of the students
spend 1-2 hours to "play or talk with friends"
89
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"do jobs at home," while 65-75% of the students
"read a book for enjoyment," and "do homework."
Table
16. Pupils'
activities
before
1 . 1 w a tc h T V
2 . I p la y
o r t a lk
w ith
f r ie n d s
3 .1 d o jo b s a t h o m e
4 . 1 p la y s p o r ts
5 . I rea d
a
b o ok
fo r
e n io y m e n t
6 . 1 u se c o m p u ter
7 . 1 d o h o m e w o rk
Education
Children
's Final
Report
spent less than two hours to "play sports,
or after
school
on a normal school
M
L e s s th a n 1
N o t im e
ofMathematics
1 -2 h o u r s
h o ur
o r e th a n 2
b u t le s s t h a n
day (%)
4 or
4 h o u rs
m o re
h ours
3 9 .8
2 5 .3
2 1 .7
6 .0
7 .2
0 .0
2 4 .1
4 5 .8
1 3 .3
1 6 .9
2 .4
2 5 .3
4 4 .6
1 8 .1
9 .6
8 .4
2 6 .5
4 1 .0
1 9 .3
4 .8
2 7 .7
4 8 .2
1 3 .3
2 .4
8 .4
8 9 .2
6 .0
2 .4
0 .0
2 .4
1 9 .3
6 .0
2 .4
7.2
6 5 .1
[6] Number of family members including
students
themselves
According to the survey, a majority of the students in both urban and rural schools have 4-9
members in their families.
However, students from the rural school tended to have more
members than those from the urban schools.
Table
17. Distribution
Family
member
of students'
Pupils'
Combined
8.4
2-3 persons
4-6 persons
7-9 persons
1 0-12 persons
More than 12 persons
(4)
Results
of Achievement
31.3
37.3
14.5
8.4
family
member
Distribution
Urban
Rural
12.0
34.0
38.0
10.0
6.0
27.3
36.4
21.2
12.1
3.0
Test
Unlike what was done for the questionnaire,
only the instructions
for the achievement test were
explained
to the students in their local language of teaching
at the beginning
of the test.
Thereafter, no further explanations
were provided to the students regarding the test in order to
avoid providing
any clues to the answers.
[1] Students'
achievements
according
to location
and sex
The average score in the mathematics achievement test in Zambia was 12%. An examination of
the average scores according to the categories reveal that they are also rather low regardless of
location and sex. In this sense, it may not have much meaning to make a comparison across
categories;
however, if we dare do so, it may generally
be stated that the difference
between
location (urban vs. rural) is much larger than that between sexes (male vs. female), as shown in
the table below.
In detail, students in the urban school were able to achieve almost twice more than those in the
rural school. According to sex, girl students
performed better than boy students, which was
90
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^____
against
ofMathematics
Education
Children 's Final Report
our assumption.
Table
18. Distribution
of students'
right
answer
(%)
C o m b in e d
M a le
F e m a le
H ig h e st
L o w e st
C o m b in e d
1 2 .0
l l .2
1 2 .9
4 5 .2
0 .0
U rb an sc h o o l
1 5 .1
14 .4
1 6 .0
4 5 .2
0 .0
R u ra l sc h o o l
8 .4
6 .5
1 0 .0
17 .8
0 .0
[2] Analysis
of the results
with regard to content domains
Similar
to previous results in (1), the achievement
of students is rather low across content
domains. If the results are examined with regard to the type of questions,
almost none of the
students were able to answer the short-answer questions.
With regard to multiple-choice
questions,
it appears that students were able to achieve slightly
better in "measurement" and
"algebra" in multiple-choice
questions, although only a few students were responsible
for this.
T a b le 1 9 . A v e r a g e r a t e s o f c o r r e c t a n s w e rs b y c o n t e n t d o m a in
^
^
^
H
N u m b er
M
A
D
P
e a su re m en t
lg e b r a
a ta
a t e rn s , r e la tio n s
a n d fu n c tio n s
G e o m e t ry
T o ta l
M u lt ip le - c h o ic e
N o .o f
q u e s t io n s
S h o rt- a n s w e r
N o .o f
R a te s o f
q u e s tio n s
c o r re c t
R a te s o f
c o rrec t
19
12
a n s w e rs
(% )
1 6 .0
2 0 .5
4
4
5
1 9 .1
1 6 .8
1 6 .0
9
4
1
4
1
4
1 2 .4
1 7 .2
6
25
48
N o .o f
q u e s t io n s
a n s w e rs
(% )
3 .4
0 .0
28
0 .0
0 .3
1 .2
3 .7
2 .2
U rb a n sc h o o l
M u lt ip le c h o ic e
S h o r t-
c o r re c t
a n s w e rs
(% )
Table 20. Average rates of correct answers of content
T o ta l
R a te s o f
16
5
8
6
1 2 .0
1 5 .4
1 5 .3
8 .5
1 3 .5
10
73
7 .2
1 2 .0
domain
R u ra l s ch o o l
T o ta l
M u lt ip le -
S h o r t-
c h o ic e
T o ta l
N um ber
2 0 .1
a n sw e r
5 .1
1 5 .3
l l .2
a n sw e r
1 .4
8 .1
M e a su re m e n t
2 4 .1
0 .0
1 8 .1
1 6 .2
0 .0
1 2 .2
A lg e b ra
2 1 :7
0 .0
1 7 .4
1 6 .0
0 .0
1 2 .8
D ata
1 9 .0
0 .5
9 .8
14 .1
0 .0
7 .1
P a tte rn s , r e la tio n s
2 1 .3
2 .2
1 8 .1
9 .7
0 .0
8 .1
a n d f u n c tio n s
G e o m e tr y
1 7 .4
6 .9
l l .1
6 .4
0 .0
2 .6
There was one aspect that was found strange in the multiple-choice
questions. In a majority
the questions in the achievement test, 4 choices were provided. This implies that the rate
correct answers could be around 25%, even if students guess their answers. However, in all
domains, the rates were lower than 25%. A probable explanation
for this is that almost 30%
91
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of
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of
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the students, on an average, either
meaningful.
did not provide
of Mathematics
Education
Children
any answers at all or their
's Final
Report
answers were not
Discussion
From the performance of the students in this survey, it can be said that a majority of the students
were neither ready nor sufficiently
competent to take this type of achievement test. With regard
to the results from the questionnaire,
certain factors from the viewpoints of either mathematics
education or others must be indicated.
In Zambian case factors in external subject education
appear to affect students' actions more seriously. Prominent among them are as follows.
[1] Language
Language appeared to be one of the biggest factors that resulted
in the low performances of
students. As revealed by the questionnaire,
over 90% of the students do not speak English
frequently
at home. Besides, reading comprehension
in English
was not sufficiently
good to
understand the level of language used in both the questionnaire
and test. However, it must not
be concluded that only their mother tongue must be used due to a number of reasons.
For example, a Zambian scholar expressed that there has been no attempt thus far to interpret
the contents of mathematics to local languages in the written form; the contents are written in
English. It was analyzed that this fact may have resulted in a certain understanding
gap between
learning in English and other mother tongues.
A second example may be with regard to the level of development in reading. While observing a
lesson in one of the schools, there were several students who copied what the teacher was
writing on the blackboard
into a series of letters without spaces, which did not form words. This
may reveal that such students did not obtain any meaning from the sentences written on the
board. In such a case, there would not be much difference in giving written tests either in the
local languages or English.
[2] Students'
experience
or preparedness
for writing
the exam
It appears that a majority of the students at this level were not used to taking tests that had an
enormous volume of question papers, which were given to individual
students. They were also
not familiar with questions such as multiple-choice,
sentence problems, open-ended questions,
etc. In addition
to the above, it was found that there are a number of skills required
either
explicitly
or implicitly
for taking exams. These could be understanding
the multiple-choice
answers in the context of the questions,
matching the answer numbers to the actual written
answers, recognizing
the answering spaces for open-ended questions,
deciphering
the meaning
of pictures in the context of the questions, etc, to mention a few.
A little more care must be given to knowing the extent that the students have developed
skills in writing exams and how far they can be trained in the targeted grade.
[3] Existence
of silent
such
responders
According to the findings
in 2.2, there were a number of students who did not provide
answers or those that were meaningful;
this made the results less credible
for ascertaining
difficult,
easy, or favorite topics for the students in Zambia.
any
the
On the other hand, this fact can be utilized
for analyzing students' tendencies
or preferences
answering questions by comparing the rates of responses with other findings
in this survey.
For example, there is a large variation
in the rates of students'
responses toward answering
questions,
which varies from 87% to 6% on the opposite ends of the spectrum. By comparing
92
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of Mathematics
Education
Children
's Final
Report
the type of questions
according to the rates of responses with previous tests conducted by the
school, the manner of solving questions,
topics in the subject,
a trend with regard to the
experience of students with tests, their cognitive
power, or preference of topics in mathematics
may be ascertained.
Taking into consideration
the abovementioned
aspects, it is important
to examine further the
hindrances
or perceptions
that may impede the ability
of the students to tackle questions
not
only from the viewpoint
of mathematics education but also from the cognitive,
cultural,
and
social viewpoints.
This will, at a later stage, provide this research project with a few ideas to
alter the current approach and adapt one that is more appropriate
for ascertaining
the real
education scenario in countries such as Zambia.
2.6.4
Results
(1)
Survey
from the Second
Year Field
Schedule
The survey was conducted
table21.
during
the period
October
Table 21. Schedule
D ate
16 w / O c t / 2 0 0 5
17 m / O c t / 2 0 0 5
1 8 - 2 1 s7 O c t / 2 0 0 5
2 3 rd / O ct / 2 0 0 5
(2)
Survey
A
D
D
D
17-21,
2004;
the details
are provided
in
of data collection
A c tiv ity
rriv a l to L u s ak a , Z a m b ia
isc u ssio n w ith M r. N k h ata a t U n iv e rs ity o f Z am b ia
ata c o lle c tio n in th re e sa m p le sc h o o ls
ep a rt u re fr o m L u sa k a , Z a m b ia
Until now,Urban School and Rural school A were selected as the target average schools. At this
point, a comparison between the result of tests conducted with and without an explanation to the
students in the local language will be provided. Rural School A has only one class in each grade;
thus, an additional
rural school must be included to make a comparison. Therefore, Rural
School B was selected as an average rural school, located in Mazabuka District in the Southern
Province of Zambia.
(Explanation
in local language implies a translation, into the local language, of questions that
are written in English, and an explanation regarding the manner in which to write the answers,
particularly
in the case of multiple-choice questions.)
Table 22. Location
R u ra l P rim a ry S c h o o l A
R u ra l P rim a ry S c h o o l B
of schools
S ch o o l lo c atio n
2 k m fro m th e c a p ita l c ity , L u sa k a
5 8 k m fro m th e c ap ita l c ity , L u sa k a
1 3 k m fro m D istrict C en tre
2 4 k m fro m D is trict C en tre
In the urban school, the test was conducted in 3 classes; however, it was observed that in one
class, a teacher who was supervising the students provided them with a few answers.Therefore,
the data pertaining to this particular class has not been included in the sample. Despite this,
there were 2 classes out of which one was provided with an explanation of the test and the other
93
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Development
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Education
Children
's Final Report
was not. Rural school B had 2 classes, although these were combined and one teacher taught
both classes at the same time. In spite of this, one class was provided with an explanation of the
test in local language and the other class was not. Rural school A had just one class, and the test
was conducted with an explanation.
Table
23. Distribution
of sample
G ra d e
4
R u r a l P r im a ry S c h o o l A
R u r a l P r im a ry S c h o o l B
T o ta l
B oy s
34
19
37
90
4
4
G ir ls
35
20
12
67
T o ta l
69
39
49
15 7
The achievement test included topics that students were expected to know in Grades 3-7, as
shown in table 24. However, there were 2 questions that were not covered in the mathematics
curriculum for Grades 1-7 in Zambia.
Table
Q
Q
Q
Q
Q
Q
l (l ), Q
2,
4 ( 1), Q
5 ( l), Q
8
9 and Q
(3)
24. Curriculum
coverage
of the achievement
Q u e stio n n o .
l (2 ), Q 3 ( l), Q 3 (2 ) an d Q 4 (2 )
test
G ra d e w h e n c o v e re d
G ra d e 3
G ra d e 4
G ra d e 5
G ra d e 6
G ra d e 7
N o t c o v e re d in a iry
4 (3 ), Q 6 ( 1), Q 6 (2 ), Q 6 (3 ) a n d Q 7
5 (2 ) a n d Q 5 (3 )
lO
The performance of the students who took the test is presented
overall performance, followed by the question-wise performance.
[ 1 ] Overall
below, beginning
with the
performance
Table 25 below shows the overall performance of students who took the test.
Table 25. Pupils'
general
performance
A v e ra g e
M a le
f e m a le
E x p la n a tio n
U rb an ar e a
l l .2 %
1 2 .1 %
1 0 .4 %
1 5 .6 %
W
8 .2 %
ith o u t
R u ra l a re a
6 .9 %
5 .8 %
9 .0 %
8 .7 %
2 .7 %
A ll
8 .8 %
8 .2 %
9 .7 %
1 2 .3 %
6 .2 %
As can be seen from the above table, in the urban school, boys performed better than girls.
There was also a large difference
within the urban area between the performance of students
who received explanation
and those who did not-those
who received the explanation
performed better than those who did not. However, in the case of students in the rural area, girls
performed much better than boys; however, even in this case the performance of those who
received explanations
was superior to those who did not. It is also evident that students in the
urban school performed better than those in the rural one.
94
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[2] Question-wise
Project
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Education
Children
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Report
performance of students
Table 26 shows the question-wise
performance
of the students
in the achievement
test.
T a b le 2 6 . Q u e s t io n b y q u e s t io n p e rf o r m a n c e o f p u p ils
T o ta l
W it h
e x p la n a tio n
W ith o u t
e x p la n a t io n
T o ta l
W ith
e x p la n a tio n
W ith o u t
e x p la n a tio n
Q l ( i)
0%
0%
0%
2 6 .1%
4 2 .3 %
1 7 .4 %
(2 )
1 .5 %
3 .6 %
0%
2 0 .5 %
3 0 .8 %
1 7 .4 %
0 2
1 .8 %
7 .1%
1 .2 %
1.1%
0%
0%
Q 3 ( l)
2 4 .7 %
4 2 .9 %
1 2 .2 %
9 .7 %
1 0 .9 %
2 .2 %
Q 3 (2 )
1 6 .7 %
2 0 .8 %
1 3 .8 %
8 .7 %
1 .3 %
8 .7 %
Q 4 ( l)
4 7 .8 %
6 4 .3 %
3 6 .6 %
1 5 .1 %
1 8 .2 %
0%
Q 4 (2 )
1 .5 %
3 .6 %
0%
1 .4 %
0%
0%
Q 4 (3 )
0 .4 %
0 .9 %
0%
1 .1%
0%
2 .2 %
Q 5 (l)
4 2 .8 %
5 7 .1 %
3 2 .9 %
2 6 .1 %
5 0 .0 %
0%
Q 5 (2 )
2 0 .3 %
1 7 .9 %
2 2 .0 %
1 0 .2 %
1 9 .2 %
4 .4 %
Q 5 (3 )
8 .0 %
5 .4 %
1 0 .0 %
1 0 .2 %
1 5 .4 %
4 .4 %
Q 6 ( l)
0%
0%
0%
4 .3 %
1 0 .6 %
2 .2 %
Q 6 (2 )
0%
0%
0%
1 .0 %
3 .4 %
0%
Q 6 (3 )
0%
0%
0%
3 .4 %
5 .8 %
2 .2 %
0 7
1 2 .3 %
3 0 .6 %
0%
0%
0%
0%
Q 8
10 .1%
0%
2 2 .8 %
0%
0%
0%
0 9
2 1 .7 %
2 8 .6 %
17 .1%
1 .4 %
5 .8 %
0%
Q lO
10 .1%
2 5 .0 %
0%
1 .1 %
0%
0%
T o ta l
l l .2 %
1 5 .6 %
8 .2 %
6 .9 %
8 .7 %
2 .7 %
(There are 2 schools in the rural area and "total" includes both; however, "with explanation"
"without explanation"
are the resultsjust
in one school that has 2 classes to compare with.)
[3] Levels
of inability
of students
in solving
problems
assigned
according
to school
and
location
After taking the test, students were also interviewed
in order to identify
the levels of their
inability
to solve the problems assigned to them. The inabilities
were identified
and grouped
into 3 levels as follows:
unable to read the question
presented,
unable to understand
the
question/concept
in the question, and unable to carry out the process offinding
the answer to the
problem presented. The frequency of these inabilities
in the two categories
of schools for
questions 5 (3), 6 (1), and 8 are presented in table 27.
It is evident from the above table that a large number of students in both the urban and rural
schools were able to read the question
presented
to them. However, very few students
demonstrated
an understanding
of the question, i.e., they were not able to translate the content
of the question into the local language. With regard to working out (process) the solution to the
problem given, merely 2 students in the urban school demonstrated
mastery of the process for
part 3 of question 5. The remaining students were not able to work out the solution. Table 8 also
shows that merely 3 students provided the correct answer for part 1 of question 6. Although
certain students (numbers in brackets) provided correct answers for part 3 of question 5 and for
question 8, they did this accidentally
without using the correct processes.
95
International
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Project
Table 27. Pupils'
ocs
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3
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Q 5 (3 )
Q 6 ( l)
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27
[4] Levels of ability
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43
4
0
0
0
2
1
3
0
0
0 ( 1)
0(1)
0 (2 )
3
3
for solving
levels of ability
6
0
problems according
4 3
4 4
Table
to their performance
for solving
questions
according
errors
to their
of problem
performance
solving
to their performance.
level
N o . o f s tu d e n t s
I I . U n Cd eo rns ctae np dt in g o f
4 4
2 5
0 (2 )
4
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28. The students'
according
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of students
Table 28 shows students'
I II . P ro c e s s
44
44
Table
given
27
17
to solve problems
's Final Report
N o . o f p u p ils w it h
27
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of inability
of Mathematics
F r e q u e n c y o f o c c u rr e n c e o f in a b ilit ie s
I . R e a d in g
o
z
levels
Development
I II . P ro c e s s
w ith C o r r e c t
A n sw e r
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0
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0
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3
0
3
0
0 (2 )
0
0 (2 )
4
6
29. Pupils'
6
0
mistakes
Q 5 (3 )
D if fic u lt w o r d s &
E x p r e s s io n s
A ny
re m a rk s
re g a r d in g p r o b le m
s o lv in g p r o c e ss
G r a te r th a n
1 /4 , 1 /3
1 /4 is g ra t e r th a n
1 /3 , s in c e 4 is
g ra te r th a n 3 .
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in problem
R ura l S ch o o l
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solving
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0 (2 )
in Q5 (3)
H ig h p e r fo r m a n c e
L o w p e r fo rm a n c e
T h e y c o u ld n o t
G ra te r th a n
G r a te r th a n
re ad an y w ord s.
1 /4 , 1 /3
1/4 , 1/3
1 /4 is g r at e r th a n
1 /4 is g r a te r th a n
1 /4 is g ra te r th a n
1 /3 , s in c e 4 is 1 /3 , s in c e 4 is 1 /3 , s in c e 4 is
g r a te r th a n 3 .
1 /4 is g r a te r th a n
1 /4 is g r a te r th a n 1 /4 is g ra t e r th a n
1 /3 , s in c e 1 + 4 = 5 , 1 /3 , s in c e 1 + 4 = 5 , 1 /3 , s in c e 1 + 4 = 5 ,
1 + 3 = 4 a n d 5 is 1 + 3 = 4 a n d 5 is 1 + 3 = 4 a n d 5 is
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Table
Q 6 (l)
30. Pupils'
mistakes
U rb an sch o o l
E x p re s sio n s
A ny
rem ark s
r e g a rd in g p r o b le m
c o n ta in e r
Development
in problem
of Mathematics
solving
R u ral S ch o o l
c o n ta in e r
A ny
c o n ta in e r
T h ey
c o u ld
n ot
2 /5 + 1/ 5 = 3 / 1 0
2 /5 + 1 /5 = 3 / 1 0
2 /5 + 1/5 = 3 / 1 0
2 /5 + 1/ 5 = 2 + 5 + 1 + 5
= 13
2 /5 + 1 /5 = 2 + 5 + 1 + 5
= 13
2 /5 + 1 /5 = 2 + 5 + 1 + 5
= 13
2 /5 + 1 /5 = 2 + 5 + 1 + 5
= 13
31. Pupils'
mistakes
U rb an sch o o l
rem ark s
L o w p e rf o rm a n c e
rea d a n y w o rd s.
2 /5 + 1 /5 = 3 / 1 0
Table
E x p re s sio n s
's Final Report
in Q6 (1)
H ig h p e r fo r m a n c e
A majority of the students did not understand what the question
what needed to be done and added the numbers in the question.
0 8
Education
Children
F r a c t io n
R e la tio n s h ip
B o u gh t
b e tw e e n
5 + 3= 8
in problem
R u ral S ch o o l
F ra c tio n
R e la tio n sh ip
B o u gh t
b e tw e e n
5+ 3= 8
implied
solving
so they merely guessed
in Q8
F r a c tio n
R e la tio n sh ip
B ought
B e tw e e n
5+3= 8
r e g a rd in g p r o b le m
L o w p e rf o r m a n c e
T h e y c o u ld
no t
r e a d a n y w o rd s .
M o s t o f th e m
not
a n y th in g
d id
a n sw e r
All the students did not understand the meaning of the question;
therefore, it was explained
in
the local language. Then, they did understand the situation, which was that "Jody bought 5 kg of
meat and George bought 3 kg of meat"; however, they did not understand what "the relationship
between Judy's and George's meat in terms of weight" implied.
Discussion
1. Analysis
of the Results
of the Students'
Achievement
Test
Ql (1) was an easy addition
of fractions:
2/5 + 1/5. However, no students in the urban school
answered correctly, and a large number of them wrote 3/10 as the answer. Thirteen students
from among 157 added all the numbers that are shown there; this implies that they did 2 + 5 + 1
+ 5 and got 13. Eighteen
of them wrote 310 as the answer (3 may have been obtained from 2 +
1-the sum of the numerators, and 10 may have been obtained from 5 + 5-the sum of the
denominators).
Ql (2) was a subtraction
of two fractions:
5/7 - 2/7. Thirty-one
students provided
3/0 and 16 of them answered 21, which is the sum of all the numbers there.
the answer
Merely 3 students provided the correct answer (3/7) to Q2 (Express the answer in a fraction-3
h-7); 52 students wrote 2 or 2 remainder 1, which is the quotient of7 å *•E
3.
With regard to sentence questions, a majority of the students appear to be unable to
understand them well; this was true even in the classes that were provided explanation
local language because a large number of students merely added the numbers provided
questions,
although they were not questions of addition.
In the case ofQ6 (3), the answer
and in the case ofQ8, itwas 8.
97
read or
in the
in the
was 7,
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A large number of students did not understand how to answer or what they were asked, so they
wrote the answer at an inappropriate
place, or what they wrote was incomprehensible.
In terms of the comparison between results of those students who were provided an explanation
in the local language and those that were not, there were rather large differences.
The greatest
difference was that the students who were provided explanation
in the local language knew how
to write answers, particularly
for multiple-choice
questions;
on the other hand, a majority of the
students who were not provided any explanation
failed to write the proper form of answers in
the adequate place.
(4)
Analysis
of the Results of the Interview
It was rather difficult
to communicate with the students in the interview because they did not
understand English
very well. A large number of students failed to read the questions
aloud;
certain students were unable to read even a single word. Students in the urban school could read
more fluently as compared with students from the rural school.
Q5 (3) Which ofthefollowing
the parenthes es.
1. A is greater
A:1/4
B:1/3
is greater
than B.
than the other?
2. B is greater
Answer(
than A
Write the correct choice (1, 2, or 3) in
3. A = B
)
A majority of the students believed
that 1/4 was greater than 1/3 since 4 is greater than 3. They
did not recognize that a fraction is a number, butjust
looked at each number shown there.
Q6(l) When you add 2/51 of water to 1/51 of water in a container,
the container?
how much water is there
in
A large number of students were unable to read the question;
however, they guessed what it
stated and believed that the question asked for the 2 fractions to be added. Certain students did
not even have a vague understanding
of what they were supposed to do. When the meaning of
the question was explained
in local language they were able to understand it. However they
were unable to add the fractions.
Several students added not only the numerators, but also the
denominators.
Others calculated
in the following
manner: 2/5 + 1/5 = 2 + 5 + 1 + 5 = 13. Merely
3 students provided the correct answer.
When we asked the students to write the process of calculation
on paper, several students were
unable to use mathematical
symbols like +, -, =,etc.; thus, the numerals appeared as 2 5 1 5 13
on the paper.
Q8 Judy bought 5 kg of meat. George bought 3 kg of meat. Write afraction
in the bracket
between Judy 's meat and George 's meat in terms of weight.
George'smeatis
(
to
) ofJudy'smeat.
None of the students understood the implication
of the above question. When it was explained
in the local language they were able to understand the quantity of meat that Judy and George
bought; however, they were unable to understand what "relation" meant. They had not learned
98
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fractions
knowledge.
in terms
Project
of proportions,
and this
question
ofMathematics
Education
Children
was too advanced
's Final Report
for their
current
In addition,
certain students who provided the correct answer on the test did not answer
correctly
in the interview.
It was possible
that they were upset because they had not been
interviewed
in this manner before.
Conclusion
Several students calculated
a fraction in the following
manner: 1/2 = 1 + 2 = 3. This implies that
students do not consider a fraction to be a number but a combination
of two numbers that is
supposed to be added.
Further, a majority of them do not understand English very well. They may be used to attending
class without studying, where they can merely guess what the teachers are saying.
Future research must be conducted in order to explore what
understanding
of students with regard to mathematical
concepts.
is required
to improve
the
Acknowledgment
We greatly appreciate
University
of Zambia,
the help
in writing
rendered by Prof.
this report.
Chistopher
Haambokoma, professor
Living
In Zambia - 1998, 1 8.
Referen ces
Central
Statistical
Curriculum
Office, (1999).
Development
Centre, (2003).
Conditions
Mathematics
99
Syllabus
Grade 1- 7.
in the
International
2.7
Cooperation
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Education
Children
's Final Report
Japan
CHIKARA KINONE
2.7.1
School
Curriculum
In the elementary schools of Japan, the topic "Fractions" is taught in Grades 4, 5, and 6. Table 1
presents the outline of the content development of fractions in the abovementioned grades.
T a b le 1 . o u t lin e o f C o n te n t d e v e lo p m e n t o f fra c t io n s in th e C o u rs e o f S t u d y f o r
E le m e n ta ry S c h o o l
T erm s a n d sy m b o ls
C on te nts
G rad e
4
> T h e m e an in g o f frac tion s
> H o w to ex p ress fra ction s
> F ractio n s can b e re p re sented as a c erta in n u m b er o f tim e s u n it
fr a ctio n s
>
>
5
6
2.7.2
F ractio n s o f th e sa m e s iz e
C o n v ertin g w h o le n u m b ers an d d ecim als in to fr actio n s, an d
rep re sen tin g fra ctio n s in d ec im a ls
> T h e resu lt o f d iv id in g w h o le nu m b ers c an alw ay s b e ex p ressed
a s a sin g le nu m b er u sin g fr actio n s
> A d d in g an d sub tra ctin g fractio n s w ith th e sam e d en o m in ato rs
> T h e e q u iv a len ce an d size o f fr actio n s, a n d th e m eth o d s h o w to
c o m p are th e ir siz e
> A d d in g an d su b tra ctin g fractio n s w ith d ifferen t d en o m in ato rs
> M u ltip ly in g an d d iv id in g fr actio n s
>
>
>
>
>
D e n o m in ato r
N u m e rato r
M ix e d fractio n
P rop er fr a ction
Im p ro p er fr actio n
(N oth in g )
> R ed u ctio n to a
com m on
d e n o m in ato r
Basic Information
(1)
Schooling
Age
The school education system in Japan comprises elementary education for 6 years, junior
secondary education for 3 years and senior secondary education for 3 years (6-3-3 system).
Further, Japanese students enter elementary school at the age of 6, junior secondary school at
the age of 12, and senior secondary school at the age of 15.
Table 2. Schooling
A g e : 6 - 1 2 (fo r 6 y e a r s )
E le m e n ta ry S c h o o l
(2)
School
Calendar
age in Japan
A g e : 1 2 - 1 5 (fo r 3 y e a r s)
J u n io r S e c o n d a ry
A g e : 1 5 - 1 8 (fo r 3 y e a rs )
S e n io r S e c o n d a r y
and Examination
Generally
speaking, a school year in Japan begins in the second week of April and ends in the
fourth week of March. It is divided
into 3 terms; there are school holidays
after the completion
of 2 terms, as shown in Table 3. (Recently,
certain schools have begun introducing
a 2-term
system.)
100
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T a b le 3 . S c h o o l c a le n d a r in (y e a r)
J an .
F eb .
T erm 3
M a r.
A p ril
M ay
Ju n e
T erm 1
Ju l¥
A ug.
S ep .
O ct.
N ov.
T erm 2
Il
D e c.
At the elementary school level in Japan, the end-of-term test or national examination is not
conducted as a legal requirement. However, a type of evaluation test is conducted in order to
assess students' achievement depending on each school or teacher.
(3)
Promotion
System
In Japan, elementary and junior secondary education is compulsory for a total of 9 years. All
students can be automatically
promoted during this period. This implies that all students in each
class of elementary and junior secondary school in Japan are approximately
the same age. After
junior secondary school, there is an entrance examination for senior high school; students who
want to enter senior high school must pass this examination.
The recent enrolment ratios of elementary
senior high school are as follows:
Table
4. Enrolment
high
and Progression
S ch o o l
E n r o lm e n t o r P ro g r e s s io n R a tio
E le m e n ta ry
9 9 .9 6 %
Source:
(4)
and junior
schools
ratio
and the progression
in Japan
ratio of
(2005)
J u n io r S e c o n d a r y
S e n io r S e c o n d a r y
9 9 .9 8 %
9 7 .6 %
http://www.mext.go.Jp/b
menu/toukei/002/002b/1
8/0 1 5.pdf
In Japan, the official language and mother tongue for both children
general; therefore, the medium of instruction
is also Japanese.
(5)
Class
and teachers
is Japanese
in
Organization
In the elementary schools of Japan, each teacher takes charge of a particular class and teaches
all subjects (class-specific
teacher). In general, each lesson is of a duration of 45 minutes, and
there are at least 23-27 lessons per week.
Table
H ou r
G ra de
8
9
P erio d
H R
2.7.3
Results
(1)
Survey
1st
10
,nd
5. School
ll
3rd
from the Second
hours
F irst sh ift
12
13
A ll G ra de s
L u n ch T im e
4 th
C lean in g
Year Field
14
5th
15
6th
16
H R
Survey
Schedule
The survey in Japan was conducted according to the following
Table
D a te
0 6 /0 3 /2 0 0 6
6. Schedule
schedule:
of data collection
A ctiv ity
D a ta co lle ctio n in th e sch o o l, O n o m ich i-city , H iro sh im a
101
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and Samples
Onomichi city is one of the cities in Hiroshima prefecture and is located in the southeast of
Hiroshima. The population
of the city is approximately
150,000.
An average school in
Onomichi city was selected as the sample school based on an achievement test conducted by the
education board of the city.
Table
7. Location
of schools
S ch o o l lo catio n
O n th e to p o f a h ill w h ich is in O n o m ic h i-c ity
A v erag e E le m en tary S ch o o l
Grade 4 was selected as a sample class for the survey. Although the actual strength of the class
was 23 students, 4 were absent (due to illness)
when the survey was conducted. The distribution
of the sample for the survey was as follows:
Table
A v e r a g e E le m e n ta ry S c h o o l
8. Distribution
G ra d e
4
of sample
B oys
5
G irls
14
A ge
10
T o ta l
19
Due to time limitations,
the survey in Japan included only these 19 students studying
However, all 19 students participated
in the achievement test and were interviewed
NewmanProcedure by 7 graduate students of Hiroshima University.
(3)
Results
of the Achievement
Test
From the result of the achievement test, the average score and the highest
amongthe 19 students, 5 boys and 14 girls were found as follows:
Table
9. Results
A v e ra g e
A v erag e S ch o o l
6 0 .7 %
in grade 4.
using the
of the achievement
B oys
G ir ls
5 7 .4 %
6 1 .9 %
and lowest scores
test (%)
H ig h e s t
8 7 .0 %
L o w est
1 5 .0 %
Figure 1 presents the individual
achievements of the 1 9 students in decreasing
order. As a result,
J20, J01, J13, J22, and Jll were identified
as High Performers and J06, J15, J09, J18, and J17
were identified
as Low Performers for the analysis
of the interview
using the Newman
Procedure. Incidentally,
the scores of the high and low performers were over 75.0% and less
than 45.0%, respectively.
100
Figure
1. Individual
Achievements
102
of the Students
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Table 10 shows the question-wise
achievements of the students in percentage. In addition,
combined achievements are represented in decreasing order as bar graphs in Figure 2.
N o .
Q
Q
Q
Q
Q
0
0
0
Q
i
l
2
3
3
4
4
4
5
(i)
(2 )
Q
Q
Q
Q
Q
0
Q
Q
Q
5 (2 )
5 (3 )
6 ( l)
6 (2 )
6 (3 )
7
8
9
lO
(l)
(2 )
(1)
(2 )
(3 )
(l)
achievements
C o n te n t
C o m b in e d
9 4 .7 %
7 8 .9 %
5 .3 %
8 9 .5 %
7 9 .8 %
7 3 .7 %
5 2 .6 %
7 3 .7 %
9 7 .4 %
1 /5 + 2 /5
5 /7 - 2 /7
3- 7
3 / 4 m in T a p e - D ia g r a m
l / 3 m in T a p e - D ia g r a m
1 / 2 o f 2 m in T a p e -D i a g r a m
l /2 m in T a p e -D ia g r a m
1 'A m in T a p e - D i a g r a m
O r d e r in g 2 / 5 a n d 4 / 5
O
O
A
C
6
C
R
R
M
of pupils
r d e r in g 3 / 6 a n d 1 /2
r d e r in g 1 / 4 a n d 1 / 3
d d 2 / 5 C t o 1 / 5 -C
u t 2 / 7 m fr o m 5 / 7 m
p ie c e s o f l /6 m
h a n g e 0 .7 t o a fr a c t io n
e la t i o n 5 k g a n d 3 k g
e p r e s e n t in g " H a l f
a k e s e n te n c e p ro b le m
2
2
8
8
4
5
6 .3 %
6 .3 %
2 .2 %
4 .9 %
7 .4 %
0 .0 %
5 .3 %
3 1 .6 %
5 0 .0 %
the
in percentage
A v e ra g e S c h o o l
B oys
1 0 0 .0 %
8 0 .0 %
2 0 .0 %
1 0 0 .0 %
6 0 .0 %
4 0 .0 %
4 0 .0 %
8 0 .0 %
9 0 .0 %
2
6
7
7
5
5
0
0
7
5
5
0
0
30
60
.0
.0
.5
.0
.0
.0
.0
.0
.0
%
%
%
%
%
%
%
%
%
G ir l s
9 2 .9 %
7 8 .6 %
0 .0 %
8 5 .7 %
8 6 .9 %
8 5 .7 %
5 7 .1%
7 1 .4 %
1 0 0 .0 %
28
14
83
88
44
50
7
32
46
.6 %
.3 %
.9 %
.4 %
.6 %
.0 %
.1 %
.1 %
.4 %
1OO.O%
90.0%
SO. 0%
70.0%
60.0%
5O.O%
4O.O%
30.0%
2O.O%
10.0%
0.0%
Figure
(4)
2. Question-wise
Results
of Interview
achievements
using
of students
in percentage
(Combined)
Newman Procedure
Before presenting the results of the interview,
of Interview items for Students is presented.
a comment on the given framework for the results
As is evident from reports on other countries, researchers were provided a commonframework
for presenting
the results of the interview;
this framework included Frequency offailure
on
problem-solving level with its three components i.e., Reading, Understanding of Concept, and
Process. However, a difficulty was encountered in applying
the results from the interview with
Japanese students to the framework. It appeared that the interview items were not corresponding
103
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favorably
Cooperation
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with the abovementioned
of Mathematics
i)
ii)
Hi)
iv)
v)
vi)
's Final Report
components.
Therefore, this report reconsiders
the framework by comparing
steps of the Newman Procedure.
The interview
Education
Children
the interview
items with the
items were as follows:
Ask them (the students) to read to you the sentence of the question. Can they correctly
readit?
If there is any difficult term/expression for them to read, describe it below.
(Note: This item asks interviewers
to write the term/expressions
down in the
interview sheet, if any.)
Ask them to explain how they solved the question in words and/or any diagrams,
Ask them to write their numerical expression to find the answer for the question.
(Note: This item was used only for Q6 (1), not for Q5 (3) nor Q8)
Ask them to calculate the numerical expression. Can they correctly calculate it? (Note:
This item was used only for Q6 (1), not for Q5 (3) or Q8)
Describe any specific mistakes in the calculation,
if there are any. (Note: This item
asks interviewers
to write the errors down in the interview sheet, if any.)
As confirmed in Chapter
Table
1, each stage in the Newman Procedure is defined
ll.
Meaning
N ew m an Procedure
R e ading
C om preh ension
Transform ation
P ro cess skills
E n co ding
of Each Step
as follows:
in Newman Procedure
M eaning of each step
R eading the problem
C om preh endin g w hat he/she read
C arry ing out the transform ation fr om the w ords to selection
of an appropriate m athem atical 'm odel'
A pplyin g necessary process skills
E ncod ing the an sw er
Taking into consideration
the implication
of these steps, interview items i) and ii) correspond to
Reading in the Newman Procedure. If we regard the mathematical
"model" in the Newman
Procedure not only as symbolic models with mathematical
symbols (such as numbers, variables,
symbols of operation,
etc.), but also as realistic
models (such as length, mass, area, etc.),
illustrative
models (such as diagram, graph, etc.), linguistic
models (explaining
the rules of what
students have learnt in previous mathematics
lessons),
etc., items Hi) and iv) correspond to
Transformation. Further, item v) corresponds to Process skills. Item vi) was included in order
for researchers to analyze commonerrors committed by students. In addition,
the reactions of
students that correspond to Encoding can be found as the answer of each question in the
achievement test. However, it appears that no interview items correspond to Comprehension,
although
Understanding
of Concept in the given framework most likely
corresponds
to
Comprehension in the Newman Procedure.
Therefore,
Table 12 summarizes the correspondence
between the stages of the Newman
Procedure and the interview items.
As is evident from Table 12, instead of Reading, Understanding of Concept, and Process,
Reading, Transformation, and Process skills have been used as the components of Frequency of
failure in the problem-solving
level in the following result tables.
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Table 12. Correspondence
N ew m an Procedure
Development ofMathematics
between NewmanProcedure
for Pupils
Education
Children
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and the Interview Items
Q uestion in th e Interview Item sf or P up ils
Q 5(3)
Q 6(l)
Q8
A sk th em to read to y ou the sentence of
th e q uestion. C an they correctly read
it?
ii) If there is any difficult term /exp ression
for th em to read, describe below .
o.
o
o
o
o
o
X
o
X
o
o
o
i)
R eading
C om p rehens ion
Transform ation
P rocess skills
E ncoding
H i) A sk them to exp la in h ow they solved
th e q uestion in w ords and/or any
diagram s.
iv) A sk them to w rite their n um erical
exp ression to find the answ erfor the
q u estion.
v) A sk them to calculate the num erical
exp ression. C an th ey correctly
calculate it?
(W e can assess this stage fr om the answ er of
the achievem ent test.)
The survey in Japan covered only 1 elementary school, as mentioned above; therefore, a
comparison between urban and rural areas with regard to the students' errors in problem-solving
wasnot conducted unlike the surveys in other countries.
The survey in Japan involved all 19 students in the interview; the results of the interview with
these students are presented in Table 1 3 with the alternative framework.
Q u e st io n N o .
0 5
(3 )
0 6
(1)
Q 8
Legend:
an d
C o n te n ts
O r d e r in g 1 /4
a n d 1 /3
A d d 12/5/ 牀
5 ｣ to
R e la tio n 5 k g
an d 3 k g
level in average school
F re q u e n c y o f fa ilu r e o n P ro b le m so lv in g le v e l
/ . R e a d in g
//. T r a n sf o r m a tio n
1
12
1
2
1
17
// / . P ro c e s s s k ills
N o . o f stu d e n ts
w ith C o r re c t
A n sw e rs
5
1
15
2
The reason why the total number of the first and third rows (Q5 (3) and Q8) were not 19 is that a
pupil failed putting correct answer despite making correct transformation
in Q5 (3), and that a
pupil put correct answer despite making wrong transformations
in Q8.
As mentioned earlier, 5 high performers (J20, J01, J13, J22, and Jl 1) and 5 low performers (J06,
J15, J09, J18, and J17) were selected
according to the achievement test. Table 14 shows the
students' errors in the problem-solving
level according to their performance.
105
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errors
of problem solving
performan ce
F r e q u e n c y o f fa ilu r e o n P ro b le m
Q u e s tio n N o .
/ . R e a d in g
an d
C o n te n ts
J3
M l
X
Q 5
O rd e r in g 1 /4
(3 )
a n d 1 /3
A d d l/2 5/5｣ 1 to
Q 6
(1)
Q 8
R e la tio n 5 k g
｣
o
蠎
J
xi
60
fi
=s
o
J
level according
so lv in g le v e l
ll . T r a n sf o r m a tio n
C3
e2
J3
M
fi
1
1
1
4
5
0
1
1
0
1
1
0
1
1
4
4
8
｣
o
hJ
0
to their
C o r re c t A n s w e r s
J 3.
60
X
C3
e2
0
s Final Report
N o . o f s tu d e n ts w ith
// /. P ro c e s s s k ills
Kf
+o->
H
0
Education
Children
｣
o
蠎J
ca
+-ｻ
(2
3
0
3
5
2
7
0
0
0
0
an d 3k g
Tables 15 to 17 show the specific
groups of high and low performers.
Table
S p ec ific
m istak e s
of students
mistakes
in problem
H ig h p erform an c e
16. Pupils'
mistakes
D iffi cu lt w ord s
& -E xp ressio n s
(N o th in g )
in problem
J 13 , J2 2 :
T h e p u p ils sa id , " In th e q u estio n , th ere
w as th e w o rd "ad d ". T h en I th o u g h t
th ere w a s th e n ee d to carry ou t th e
a d d itio n 1/5+ 2 /5 . B u t th eir d en o m in ators
w ere sam e , so th at I a d d ed o n ly th e
n u m erators.
m istak e s
in Q5 (3)
JO 8 , JO 9 , J 15 , J 18 :
C o m p are d o n ly th e d en o m in ato rs.
H igh p erfo rm a n ce
S p ec ific
to the
n u m e rators.
J IL J2 0 :
A ft er ex p re ssin g
th e fra ctio n s as
d iag ram s, th e p u p ils c o m p are d th em .
J 13 :
T h e p u p il said "if th e fr a ctio n s
re p re sen ted a re a, I th in k th at 1/3 is
larg er." (In fact, h e g o t w ro n g an sw er.)
J 13 , J2 2 :
C o m p ared on ly th e d en o m in ators.
Q 6 (l)
reg a rd in g
p ro b lem so lv in g
p ro c ess
solving
according
J O l:
A ft er ch an g in g th e fr a ctio n s in to
eq u iv a le nt
ones
w ith
th e
sa m e
d en o m in ato rs, th e p u p il co m p are d th e
(N oth in g )
Table
A n y rem ark s
from the interview
JO 6 . J 15
C h o ice
JO 9 :
T h e p u p il sa id , "if 1/3 is le n g th , it is
lo n g er th an 1/4 . B u t if nu m b er, 1/3 is
sm a lle r th a n 1/4 ."
Q 5 (3 )
D ifficu lt w ord s
& E x p ressio n s
A n y rem ark s
reg ard in g
p ro b lem
15. Pupils'
responses
solving
in Q6 (1)
L ow p erfo rm an c e
JO 6 . J 18 :
P ro c ess to g et th e c o rrect an sw e r
JO 6 :
C o n tain er
JO 8 :
T h e p u p ils sa id , " I a d de d 1/5 an d 2 /5 .
B u t th ere is n o n ee d to ch an g e
d en o m in ato rs , so th at I ad d ed o n ly th e
n u m e rato rs.
J 18 :
A d d in g tw o n u m erato rs,
d en o m in ato rs.
(N oth in g )
106
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tw o
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Table
17. Pupils'
Q (8)
mistakes
in problem
A n y rem ark s
reg a rd in g
pro b le m so lv in g
p ro c ess
Sp ec ific
m istak es
(N o th in g )
JO L J 13 :
T h e p u p ils g u essed th e an sw e rs " 3 /5 "
b ase d on th e w o rd "fractio n " an d th e
n um b ers in th e sen te n ce , b eliev in g th at
th e n u m erator o f a fr actio n is sup p o se d to
b e sm a lle r th an th e d en o m in ato r.
J O l, J I3 , J2 0 . J2 2 :
A tta ch e d th e u n it " k g " to th e ir an sw er
" 3 /5 " .
2.7.4
Discussion
(1)
Analysis
of the Results
from Pupils'
's Final Report
in Q (8)
L o w p erfo rm an ce
JO 6 :
kg
JO 9 :
R elatio n sh ip in te rm s o f w e ig h t
JO 8 , JO 9 . J 15 . J 18 :
T h e p u p ils g u esse d th e a n sw ers " 2 ",
"8 /5 " " 1/8 " o r " 1/2 " b ased o n th e w o rd
"fr ac tio n " an d th e n u m b ers in th e
se n ten c e, su b tractin g 3 fr o m 5 , ad d in g 3
an d 5 , arran g in g n u m b ers an d so o n .
D iffi cu lt
w ord s &
E x p res sio n s
solving
Education
Children
Achievement
Test
Prior to analyzing the results of the achievement test, each question in the achievement test is
compared with Table 1 in order to identify which questions students had already learnt and
which they had not learnt at the time of conducting the test. As can be seen from Table 18,
Questions 3, 4, and 9 had been taught previously;
however, the other questions (except Question
1 0) were not previously taught.
Table
N o
18. Questions'
Q u e st io n
Q i (D
(2 )
Q 2
1 /5 + 2 /5
5 /7 - 2 /7
3 -7
m m m T a p e a a g5f a im
IIu '3 m iii
w aaI
H1
IB p e -D iim
蝣B
T a p e - la g r a im
i H I!I -4 m in iT a p e 'm snFa im
Q 5 (l)
O rd e r in g 2 /5 a n d 4 /5
Legend:
X: It is not covered;
a
the shaded
At this point, the results of the
previously
taught and those
percentages of correct answers
on the questions not previously
C o v erag e
in G r a d e
G rad e 5
G rad e 5
G rad e 5
<3
m
m I
i
<3
G r
i
G rad e 6
questions
Coverage
in Grade
N o
0
Q
0
Q
Q
Q
0
5
5
6
6
6
7
8
Q u e s t io n
(2 )
(3 )
(1)
(2 )
(3 )
O
O
A
C
6
C
R
rd e r in g 3 /6 a n d 1 /2
rd e r in g 1/4 a n d 1 /3
d d 2 /5 牀 to l /5 t
u t 2 /7 m fr o m 5 /7 m
p ie c e s o f 1 /6 m
h a n g e 0 .7 to a fr a c tio n
e la t io n 5 k g a n d 3 k g
C o v e ra g e
in G ra d e
G ra d e 5
G ra d e 6
G ra d e 5
G ra d e 5
G ra d e 5
G ra d e 5
G ra d e 6
.&;.
Q lO
are taught
M a k e s e n te n c e p ro b le m
X
already.
achievement test are indicated
again; the questions that had been
that were not have been distinguished.
Figure 3 shows the
on the questions previously taught, and Figure 4 shows the ones
taught.
107
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100.0%
100.0%
90.0%
90.0%
80.0%
70.0%
80.0%
60.0%
50.0%
60.0%
40.0%
40.0%
30.0%
30.0%
20.0%
20.0%
10.0%
10.0%
Development
of Mathematics
Education
Children
's Final Report
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70.0%
50. 0%
0.0%
0.0%
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Figure 3. Percentages
Questions Taught
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on Questions
Taught
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10
not
From Figures 3 and 4, interestingly,
it is evident that students were able to correctly answer
certain questions that had not been previously
taught, while they were unable to correctly
answer questions that had already been taught. Therefore, it cannot be said that students will be
able to solve questions that they have already been taught and will be unable to solve questions
that they have not been taught previously.
Next, in order to identify the tendencies
of students to answer different
types of questions
correctly or incorrectly,
the questions that had been taught previously
and those that were not,
and those for which the percentages of correct answers were over 70% and under 30% were
selected. Table 1 9 classifies
these questions into 4 categories.
T a b le 1 9 : C o m p a ris o n o f P u p ils ' a c h ie v e m e n ts in
ta u g h t a n d n o t ta u g h t, m o re th a n 7 0 %
M o re th an 7 0 %
Q 3 ( l)
3 /4 m in T ap e -D iag ram
8 9 .5 %
l/3 m in T ap e-D ia gram
7 9 .8%
Q u estion s Q 3 (2 )
Q 4 (1)
112 o f 2 m in T ap e-D iag ra m
7 3 .7%
T au ght
Q 4 (3 )
1 'A m in T ap e-D ia gram
7 3 .7 %
Q 5 (l)
O rd e rin g 2 /5 an d 4 /5
9 7 .4 %
Q l(l)
1/5 + 2 /5
94 .7 %
Q uestion s Q 6 (2 )
C
u
t
2/7m
r
f
o
m
5
/7m
84 .9 %
n ot tau ght
Q 6 (l)
A d d 2 /5 i to l/5 C
82 .2 %
Q l(2 )
5 /7 - 2/7
7 8 .9 %
FromTable 19, the following
a)
b)
c)
p e rc e n ta g e b e tw e e n q u e s tio n s
a n d le s s th a n 3 0 %
L e ss th an 3 0 %
Q
Q
Q
Q
5 (2 )
5 (3)
2
8
O rd erin g 3 /6 an d l/2
O rd erin g l/4 an d l/3
3- 7
R e la tio n 5 k g an d 3 k g
2 6 .3%
2 6 .3%
5 .3%
5 .3%
tendencies in the achievements of students can be indicated:
The students were able to solve questions that asked to represent lengths
expressed in fractions as tape diagrams since they had learnt such content before.
The students were able to solve questions that comprised fractions with same
denominators, although they had not learnt such content before.
The students were unable to solve questions that comprised fractions with
different denominators since they had not learnt such content before.
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The students were unable to solve questions that asked to represent the
Relationship
between two quantities
(ratio, proportion)
and quotient of division
as fractions since they had not learnt such content before.
Comparing the above tendencies with the questions that were focused on in the interview using
the NewmanProcedure, it is evident that Q6 (1), Q5 (3), and Q8 correspond to items b), c), and
d), respectively.
By analyzing the students' responses to the questions that were not previously
taught, we may be able to gain a deeper understanding
of the learning or strategies of students in
solving unknown problems.
(2)
Analysis
of the Results
from Pupils'
Responses
to the Interview
In the interview using the Newman Procedure, the focus was on three questions-Q5
and Q8-that had not previously been taught to the students.
(3), Q6 (1),
As is evident from Table 10, the percentages of correct answers in Q5 (3), Q6 (1), and Q8 were
26.3%, 82.2%, and 5.3%, respectively.
This implies that the students could solve Q6 (1) well,
while they were unable to solve Q5 (3) and Q8. According to Table 13, it can be stated that the
reason why students were unable to solve Q5 (3) and Q8 was caused by the failure of
Transformation. For example, 12 students failed in making a transformation
in Q5 (3) and 17 in
Q8. On the other hand, in Q6 (1), merely 2 students failed in making a transformation
while the
other 1 5 provided the correct answer, thereby succeeding in making a transformation.
However, rather interesting
patterns in the process of Transformation can be observed in Table
14. According to Table 14, the high performers succeeded in making a transformation
in Q5 (3)
and Q6 (1) and failed in Q8. On the other hand, the low performers succeeded in making a
transformation
in Q6 (1) and failed in Q5 (3) and Q8. This implies that Q6 (1) was solved by
both the high and low performers, while Q8 was not; Q5 (3) was solved by the high performers
and not by the low performers.
Now, in order to analyze the students'
strategies
for solving questions
that they had not
previously
learnt, the manner in which the transformation
was made for Q5 (3), Q6 (1), and Q8
is analyzed.
At this point, the idea of Nakahara (1995)
is introduced-he
studied the Representational
System in mathematics
education. He set up 5 types of Representational
Modes-Realistic,
Manipulative, Illustrative,
Linguistic,
and Symbolic-in
the system for a constructive
approach
in mathematics education as follows:
Table
20: 5 Types of Representational
M ode
R ea listic
rep rese n ta tio n
M a n ip u la tiv e
rep rese n ta tio n
Illustra tive
rep rese n ta tio n
L ing u is tic
rep rese n ta tio n
Sy m b o lic
rep rese n ta tio n
Modes (Nakahara,
1995)
M e an in g
R ep re sen tation b y u sin g re al w o rld 's situ atio n s an d rea l th in g s,
in c lu d in g ex p erim en ts w ith c on crete an d rea l th in g s.
R ep re sen tatio n b y d o in g co n c rete m an ip u lativ e activ ities, o r b y
m a n ip u latin g le arn in g aid s a n d co n c rete th in g s w h ich ar e
m o d e lled fo r le arn in g m ath e m atic s.
R ep re sen tatio n b y u sin g d ia g ra m s, fig u res, g ra p h s a n d so o n .
R ep re sen tation b y u sin g lan g u ag e (m o th er to n g u e ) or its
ab b rev iated lin g u istic rep re se n tation s.
R ep res entation b y u sin g m ath em atic al sy m b o ls su ch as nu m b e rs,
v a riab le s, sy m b o ls o f o p eration , sy m b o ls o f re latio n a n d so o n .
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Applying Nakahara's representational
low performers make a transformation
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modes as a framework, the manner in which the high and
is classified.
Table 22 shows the manner in which students
make a transformation
in Q6 (1).
In Q6 (1), 5 high performers and 2 low performers provided the correct answer. Two high
performers
(J20
and Jll)
and 1 low performer (J09)
used realistic
and illustrative
representations
as concrete models (imaging or drawing a container with water) in order to solve
the question. Further, 3 high performers (J01, J13, and J22) and 1 low performer (J08) used
linguistic
representations
expressing
rules for translating
the question
into a mathematical
expression
(regarding
the word "add" as addition)
and for calculating
addition
of fractions
(adding only numerators without changing the denominators).
In contrast, 3 low performers provided the wrong answers in Q6 (1) and 2 of them (J18 and J15)
used linguistic
representation
like J01, J13, J22, and J08. However, they committed an
error-adding
not only the numerators but also the denominators. This implies that they merely
concentrated on the word "add" without thinking of the implication
of adding two fractions.
Twopoints emerge from this observation:
i) If students have concrete images, such as realistic
or illustrative
models, they will be able to solve Q6 (1); i ) students
who use linguistic
representation,
such as rules expressed by language, will not always be able to solve Q6 (1)
unless they understand the meaning of adding fractions or have concrete images of addition
of
fractions.
Second, Table 23 presents the manner in which students make a transformation
in Q8.
All the students failed to correctly
answer Q8, although
they displayed
various linguistic
representations.
For example, since the question asked them to "write a fraction in the bracket,"
J20, J01, J13, and J22 merely arranged 2 numbers in the question without reason in order to
make a fraction; J08, J09, and J15 added or subtracted the 2 numbers merely as a guess.
The reason why they provided wrong answers is obviously
the lack of understanding
of the
meaning of the word Relationship.
The concept of Relationship
between two quantities (such as
ratio and proportion)
appeared too abstract and far from the real world and experiences of the
students;
therefore, they were unable to come up with any concrete images or real situations
related to the concept. They merely attempted to apply "rules" or "knowledge" that they
remembered or learned before without finding any appropriate
connections
between the
question and such rules and knowledge.
Third,
Table 24 presents
the manner in which students
made the transformation
in Q5 (3).
In Q5 (3), 3 high performers provided
the correct answer. Two of them (J20 and Jll) used
illustrative
representations
drawing lines or tape diagrams divided
into 3 or 4 equal parts.
Perhaps, they imaged the fractions
1/3 and 1/4 as lengths. Another student (J01) provided
a
linguistic
representation
explaining
the method for conversion into an equivalent fraction with a
commondenominator. The student appeared to possess a deep understanding
of the meaning of
fractions;
therefore, he was able to convert into any equivalent
fraction of his choice.
On the other hand, 2 high performers and 5 low performers provided the wrong answer. Six of
them provided a linguistic
representation,
such as comparing only denominators. The question
asked to compare 1/3 and 1/4; however, the students merely thought that there was a need to
compare the given "numbers (fractions)"
in any case. Then, finding the numbers 3 and 4 in
those fractions
and comparing the numbers without considering
what those denominators
expressed, they decided that "1/4 was greater than 1/3."
Through this analysis,
it is evident that the students had 2 strategies
for solving questions that
were not previously
taught. The first strategy was to translate questions
into more concrete
no
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situations
or models that were familiar to them so that they could manage to deal with the
questions. The second strategy was to apply mathematical
rules and knowledge that they had
previously
acquired (or memorized) in solving questions. However, the second strategy can be
further divided according to whether or not the students understand the rules and knowledge.
Skemp (1979)
advocated that there were various levels of understanding,
and two
Instrumental
understanding
and Relational
understanding.
According to him,
understanding implies the ability
to apply an appropriate
remembered rule to the
problem without knowing why the rule works; Relational
understanding implies
deduce specific
rules or procedures from more general mathematical
relationships
of them were
Instrumental
solution of a
the ability to
(Skemp,
1979).
Referring to Skemp's idea, applying
mathematical
rules and knowledge that the students have
understood
is related
to Relational
understanding,
and applying
mathematical
rules and
knowledge that the students have not understood is related to Instrumental understanding.
Therefore, we can categorize the students'
taught previously
into at least 3 categories,
Table
21 : Pupils'
strategies
as follows:
Strategies
for solving
for Solving
questions
that have not been
Unknown Questions
Application
Translation
(3)
of mathematical
rules
and knowledge
based
on based
on
Relation al
Instrum ental
understan ding
un derstanding
into more concrete
models
Conclusion
From the above analyses, it has become evident that it is possible
for students to solve not only
those problems that they have previously
learnt but also those that have not been previously
learnt; in other words, they possess strategies
for solving unknown problems. For example,
although
the questions
regarding
ordering,
addition,
and subtraction
of fractions with same
denominators were not previously
taught to the students,
numerous students solved such
questions
by translating
them into more concrete models or applying
acquired
rules and
knowledge.
Undoubtedly,
the students faced difficulty
in solving problems that comprised rather abstract
content, such as expressing the Relationship
between two quantities.
In such a case, the students
merely attempted to apply rules or knowledge that they did not understand.
However, considering
students'
strategies
for solving problems
and the characteristics
of
problems, it may be possible
to seek a more fruitful learning and teaching method for students
in mathematics education. Such challenges will be focused on in our future researches.
Ill
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T ra n s fo rm
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Table 23. Pupils'W ays of M aking Transform ations in Solving Q 8
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Repre
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International
Cooperation
Project
Education
Final
Annex 1: Organization
DfiSTOSpmtteftsB*
Yea r
i-a
of the School
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Source:
Japan Ministry
Annex 2: Content
Grade
4
Grade
5
of Education,
development
Culture,
Sports,
Science
of fractions
in the Course
elementary school
and Technology
(2005)
of Study for Japanese
1. Objectives
To enable children to understand the meaning of decimals and fractions and how to represent
them.
2. Contents
A. Numbers and Calculations
To enable children to understand the meaning of fractions and how to express them.
a) To use fractions in order to represent the size of reminders,
and the size of the proportion
that is formed by division
into equal parts. To know how to represent fractions.
b) To know that fractions can be represented
as a certain number of times unit fractions.
[Terms and symbols]
Denominator, Numerator, Mixed fraction, Proper fraction, Improper fraction
1. Objectives
(1 ) To enable children to deepen their understanding
of the meaning of decimals and fractions,
and how to express them.
To enable children to understand the meaning of adding and
subtracting
fractions, to examine these calculations,
and to make use of these calculations.
2. Contents
(4) To enable children
to deepen their understanding
of fractions.
To enable children
to
understand the meaning of adding and subtracting
fractions with the same denominators,
and to make use of these calculations
in appropriate
way.
a) In simple cases, to notice fractions of the same size.
b) To convert whole numbers and decimals into fractions, and represent fractions in decimals.
c) To understand that the result of dividing
whole numbers can always be expressed as a
single number using fractions.
d) To examine adding and subtracting
fractions with the same denominator, and do these
calculations.
115
International
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Grade
6
3. Points for consideration
in dealing with contents
(3) With regard to item (4) d) in "A. Numbers and Calculations",
addition
and subtraction
as
its opposite operation of two proper fractions should be included.
1. Objectives
(1) To enable children to deepen their understanding
of addition and subtraction
of fractions,
and to make appropriate
use of these. To enable children to understand the meaning of
multiplying
and dividing
fractions,
to examine methods of calculating,
and to make
appropriate
use of them.
2. Contents
(2) To enable children to deepen further their understanding
of fractions, to understand the
meaning of adding and subtracting
fractions with different
denominators,
and to make
appropriate
use of these calculations.
a) To understand that a fraction obtained by multiplying
or by dividing
the numerator and
denominator of an existing fraction by the same number, has the same size as the existing
fraction.
b) To examine the equivalence
and size of fractions, and organize the methods how to
compare their size.
c) To examine the methods of adding and subtracting
fractions with different
denominators,
and do these calculations.
(3) To enable children to understand the meaning of multiplying
and dividing
fractions, and to
do these calculations
appropriately.
a) To understand the meaning of multiplication
and division when the multipliers
and divisors
are whole numbers.
b) To understand the meaning of multiplication
and division when the multipliers
and divisors
are fractions, on the basis of an understanding
of the calculations
involved
when the
multipliers
and divisors are whole numbers and decimals.
c) To examine the methods of multiplication
and division with fractions and decimals, and do
these calculations.
[Terms and symbols]
Reduction to a commondenominator
3. Points for consideration
in dealing with contents
(2) With regard to item (2) c) in "A. Numbers and Calculations",
addition and subtraction
of
two proper fractions should be included.
(3) With regard to item (3) in "A. Numbers and Calculations",
calculations
of mixed fractions
should not be included.
(4) With regard to item (3) b) in "A. Numbers and Calculations",
simple cases should be
handled, such as where the multipliers
and divisors are fractions with a numerator of one.
References
Institute
for International
History of Japan's
developing countries
Japan
Ministry
Education
Nakahara
Cooperation
Japan International
Cooperation
Educational
Development -What implications
today-, Japan International
Cooperation
Agency.
of Education,
Culture,
at a Glance 2005.
T. ( 1 995)
Study on constructive
Skemp R.R. ( 1 979) "Goals
No.88, pp.44-49.
of learning
Sports,
approaches
and qualities
116
Science
Agency (2004)
The
can be drawn for
and Technology
in mathematics
of understanding",
education,
(2005)
Japan's
Seibun-sya.
Mathematics
Teaching,
Chapter
Discussions
3
and Future Issues
International
Cooperation
Project
Education
Final Report
Chapter
3.1
Discussions
3 Discussions
of Present
and Future
Status
of Mathematics
Issues
Education
Although
in this research robust numerical comparisons were not always possible because of the
limitations
of our samples, making comparisons
within each country and characterizing
differences
among the participating
countries were possible and valuable. The study of syllabi
and textbooks to support the analyses was done in some cases.
In the process of comparison, four categories
were created: students-related
context, students
and learning, language, and research method. Since "learning" is a focus of this report, students
and learning are the natural one to address, while "students-related
context" and "language" are
somehow related to the preconditions
of learning,
and "research method" focuses on the
methodology
to approach these aspects of mathematics education.
Here we didn't include the "teachers and teaching" category, but this has been included
in the
second part of the joint study. In that part, we tried to uncover the status ofteachers'
teaching in
mathematics. These two reports, two some extent, clarified
the status of mathematics teaching
and learning in each country.
3.1.1
Students
related
Context
The UNESCO Framework of Education
Quality
(UNESCO, 2004) emphasizes
that context
interacts with every part of the educational
process. For students-related
context we compiled
the following
list of factors that can determine the present status of students. These contexts
affect students'
learning directly
and indirectly/explicitly
and implicitly.
The extent of each
influence and the relationship
among these interrelated
factors differs in various countries.
-Teaching in school:
belief, and etc.
Learning environment
curriculum,
in school:
textbook,
supply
teacher's
competency,
of textbooks,
teaching
-Family background:
parents' educational
background,
in education,
learning resources at home.
and learning
-Sociocultural
and religious
factors:
beliefs
governance
and etc.
our focus and sample size were limited,
(1)
Curriculum:
syllabi
belief
and etc.
and value
opportunities
and management strategies,
we could
teacher's
resources,
in education,
and values in society,
Though
reports.
strategy,
economic state, parents'
Learning environment in society:
public resources available
extra classes, private tutoring and self learning, and etc.
-Political
and economical
factors: national
market demands, examination, career ladder,
teaching
for
labour
and etc.
find the following
from country
and textbooks
All the participating
countries
have
the production
of textbooks,
while
syllabi
and textbooks, the topics to
noted the uniformity of mathematics
a national curriculum, but some introduced privatization
in
in others the government produces the textbooks.
In the
be covered are more or less similar. In fact, Nebres (1988)
curriculum despite cultural diversity.
As for the target topic of this research, fractions, some countries introduce it at very early stage
such as grade 1, and others introduce
it at a relatively
later stage such as grade 4 when they
judge that students
have become mature enough. This means that there are two distinct
approaches. One of them is to introduce the topic at an early stage and take time for the students
117
International
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Education
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Report
to digest the concept. The other one is to wait for students' readiness and to deal with the topic
in a concentrated manner.
(2)
Family
background:
parents'
background,
learning
resources
at home
In some participating
countries,
there is a gap in educational
resource at home between the
urban area and the rural area (, though the quality control in a part of the result would be
necessary in a few countries).
The relationship
between educational
resource at home and
students' achievement is, however, diversified
among participating
countries. Since the material
conditions
and requirements
of "average school" also differ in each country, other research
attempt is recommended to clarify the relationship
among those factors.
(3)
Learning
opportunities
environment
for extra
in society:
public
resources
available
classes,
private tutoring
and self learning
in education,
From an informal interview and discussion
in some participating
countries, there is a gap in
educational
resource available
in society between the urban area and the rural area. Educational
resources in society include not only physical
supply for teaching-learning
materials but also
accessibility
to extra classes, private tutoring. Apart from such factors which directly
affect
students'
learning,
the frequency and the quality of opportunities
for adult to exchange
educational
information would also have an effect upon students' learning.
(4)
Political
strategies
and economical
factors:
national
in education,
and examination
governance
and
management
Though the promotion system and the role of examinations
are different
in participating
countries, this research has revealed that examinations are a very influential
factor in each
country both in relation to teaching and to learning. Teachers tend to teach only what will be
tested, and they judge students' ability
in terms of test results. Besides, some cases during the
field survey were reported where students possibly
cheated on the test questions and even where
the teachers assisted the students in solving some questions. Although these were extreme cases,
nevertheless they indicate the disproportionate
importance of test scores.
(5)
Sociocultural
and religious
factors:
diversity
of language
and culture
From country reports in Chapter 2, the diversity
with regards to language and culture is apparent
in participating
countries. We shall address the language issue separately in 3. 1.3.
3.1.2
Students
(1)
Positive
and learning
attitude
towards
mathematics,
school,
teacher,
and themselves
In all the participating
countries except for Japan, the results show the positive
attitude
of
students towards mathematics,
their school and teachers, and themselves. There is, however, a
gap between such positive attitudes and actual achievement in mathematics in this survey.
(2)
Performance
comparisons
within
each
country
Instead of comparisons between countries, comparisons within each country may help to reveal
some of the following.
Though the result should not be overgeneralized
both within each
country and within each survey area because of the limitation
in the sample size, we shall sum
up some findings for further research.
1 ) Disparities
between urban and rural education
In most participating
countries,
the overall scores in the achievement
tests often show
superiority
of urban students. Though the lack of educational
resources would be one of the
reasons which result in failure in rural areas, we can find another reality in the result by content
domains or items. Since the urban students were not always superior to the rural students in
118
International
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every content domain or item, the research revealed the inconsistent
strength and weakness by
location in each country. Here, while we should not think little of physically
worse condition
in
the rural area, the result implies the effect of teacher and teaching on learning is more crucial.
2) Gender gaps
From country reports in participating
countries,
gender in the achievement tests was not clear.
The reason is that we could not find the consistent
result in gender by countries,
locations,
content domains, and items as well as the gap between the urban and rural areas which is
mentioned above.
3) Weak Domains and items
In the result over participating
countries in the first year survey, the content domains where the
students showed lower performance are generally "Geometry", "Algebra", and "Measurement". As
for the result in the second year survey, the items with lower performance are Q2, Q3-2, Q4-2,
Q4-3, Q5, Q6-3, Q7, Q8, and Q10.
The syllabus
coverage against the contents in the achievement tests have been often argued
throughout
the research period, such that the low coverage would result in poor performance.
However, the result of each item shows the case that students can obtain the correct answer in
uncovered topics. The discussion with regards to students' weakness would make sense, only when
weimplement in-depth analysis on their mistakes as we did in Newmanprocedure.
(3)
Students'
readiness
Cognitive,
affective and social aspects of students are all key factors. In some countries the ages
of the students in the study varied greatly from the regular age of9 (10) to as old as 15. This
indicates
the existence
of grade levels being repeated,
students dropping
out of school, and
delay in entering into primary school. In this research, we put emphasis on the present degree of
achievements,
neither maturity nor curriculum vitae.
3.1.3
Language
Students in some countries were not accustomed to the test practice
in which each student is
provided
with a sheet of white paper for taking each test. Furthermore, an analysis of some
answer sheets revealed that some students didn't understand the purpose of the questions.
In
some countries sentence problems are usually avoided on tests because they tend to be very
difficult
for the pupils.
Such situation
needs to be investigated
in relation
to the language
fluency. Because of its unique position,
we shall address the language issue separately
here.
(1)
Degree
of acclimation
to sentence
problems
Students may not be accustomed to written examinations,
even in cases where the language
used is easy and relevant to the culture. So as to investigate
such cases, an interview method, so
called Newman procedure, was employed in this research. The Newman procedure allowed us
to identify
the exact levels where students were stuck in terms of solving sentence problems.
As a country report mentioned, the understanding
or the lack of understanding
with regard to a
mathematical
concept did not directly translate to their ability
in working out the mathematical
processes. Similarly,
the students' ability
with mathematical
processes did not imply that they
were able to record their answers correctly.
This may be explained
by the fact that some
students could guess the correct answers, and are affected by the level of familiarity
with the
formulation
of the questions.
While we, of course, have to be more sensitive
in wording and
arrangements of questions,
it seems that the accustomedness
to the examination-oriented
culture is presupposition
for success in school education, which should be re-examined.
119
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(2)
Medium
of instruction
and students'
Report
understanding
The language condition
is a notable precondition
for education in most participating
countries.
In some countries the language used for providing
instruction
is the same language used in daily
life, but in other countries this is not the case. The situation
can become quite complex when
there are various overlapping
languages, such as the language of an ex-colonial
government, a
national
language and local languages
(Ex.: In Accra, Ghana they use at least the three
languages
of Ga, Asante and English).
And also, language is not only an educational
and
psychological
issue, but a political
and economical issue as well.
With use of Newman procedure, we also took into consideration
the point that language
problems are classified
as A-type problems (language fluency) and B-type problems (difference
in cognitive
structure)
(Berry, 1988).
Many teachers pointed to fluency problems among their
pupils.
A few teachers also identified
conceptual
gaps between English
and local languages.
This complex situation makes it much harder for students to express their ideas freely. There is a
lot of literature
on how difficult
it is for students to attain a certain academic level and for
teachers to manage the classrooms when the language used for teaching
is different
than the
pupils'
mother tongue (e.g. Fafunwa et al. 1989).
The interview in Newman procedure reveals that the characteristic
of question reflects on the
steps in which the pupils
made mistakes. A kind of question
reminded students of the
procedural knowledge and skills, which some students could explain how to do it in English.
Some pupils who could not explain their solving process in English showed their idea in mother
tongue in the question.
Though they could not explain why, many pupils could show their
procedural
knowledge. In other kind of question,
which students were required to show the
conceptual
understanding,
it was very difficult
for students to understand
the meaning of the
question and explain their solutions
in English.
Though we can not say "B-type problems"
existed in our survey too, it implies
that there might be several kinds of interaction
between
question and the student attempting it. It also suggests that a linguistic
support and a device in
posing questions would be essential
especially
in the questions
which require understanding
of
mathematical
concept.
(3)
Cultural
appropriateness
and social implication
Language conditions
in some participating
countries
are very complex. Multi-linguistic
and
multi-cultural
situations
would demand which language(s)
is/are to be used. In some cases there
is no word in the language that can express the concept or the word has a completely different
meaning. For example, some concepts such as "minus degree" in the first year survey may not
be culturally
appropriate
if, for example, the students don't experience minus degrees in their
daily lives. And also, the second year survey used the concepts of "sharing", "fractions", "half,
"morethan and less than" and "one-third or one-over-three". Such mathematics concepts should
have been examined from the view of cultural appropriateness,
since the students might have
different meaning and usage in corresponding
words in mother tongue.
The choice of which language to use in the classroom has not only educational
implications,
but
socfal implications
as well. This is because the acquisition
of an international
language gives
students greater access to international
information and greater opportunities
for employment
abroad. As a long-term issue, cultural appropriateness
should be carefully discussed
to broaden
and deepen students' capability
in mathematics.
3.1.4
Research
method
We raised the following
three points regarding
linguistic
influences
and new types of problems.
120
the research
method:
selection
of samples,
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Selecting
samples
and combining
research
methods
Weplanned to obtain the best possible output despite the limitation
of sample size. And the first
task was to select an "average" school in order to describe mathematics education in a particular
country. This was by no means an easy task to do because the criteria
for determining
an
"average" school within each country were not uniform. All of the target countries have unique
backgrounds
including
different
socio-economic
conditions
and different
requirements
for
schools and students. For example, an average school in Thailand
has satisfied
the minimum
condition
in terms of teaching materials and school infrastructure,
but an average school in
Zambia has fewer resources by comparison.
The research members recognized the importance of qualitative
analysis
after the field survey
reports during the workshop because each report included rich and interesting
data suitable
for
further comparative studies between countries and within individual
countries. In order to view
the subjects
from different
angles and to increase the validity
of the research, we should
combine not only different
samples but we should attempt to combine different
research
methods as well. This combination
of research methods is called "triangulation".
There are a
variety of combinations
such as pre / post -lesson interviews,
second / third-party
observations
and ideal / actual lessons.
(2)
Linguistic
and cultural
influences
on achievement
As mentioned earlier, linguistic
and cultural influences on achievement tests are obvious; it
would be applicable
not only in this research project, but also in any other international
survey.
This kind of achievement tests should be followed
by clinical
interview
method such as
Newmanprocedure to identify the degree of influence from linguistic
factors. In addition,
by
analyzing
the errors of the students
and investigating
the cause for the errors, their
misconceptions
became evident. Being able to perceive the misconceptions
of students with
regard to certain mathematical
concepts will enable teachers to understand the manner in which
they should conduct their teaching.
(3)
New types
of questions
(problem
posing)
Many developing
countries are now exploring more student-centered
lessons and the promotion
of logical thinking
skills.
Problem posing was included in this context in order to see students'
ability in this direction.
3.2
Future
3.2.1
Revisiting
Issues
endogenous
development
While writing the final report we were able to identity
some important points that need to be
clarified
further. In particular, "language problems" and "students-related
context" have become
the key words for pursuing a spirit of endogenous development. When we consider these two
key words, it becomes clear that greater consideration
of the research framework is needed.
Therefore, for the short-term we compiled a list of the issues that we faced during the course of
this study. This, in the long term, will form the starting point for our future research for the
development of a comprehensive framework for mathematics education research in developing
countries.
3.2.2
Future
research
By the endeavor of this research group, we have to have a target to improve
learning in each country although we should not be too short sighted.
121
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(1 )
Way forwards
Endogenous development will continue to be a very basic presupposition
for our future research.
And this endogenous-ness
should
be considered
in curriculum,
teachers,
and students.
Endogenous development of curriculum
is based upon the society's
needs and may consider
possibility
of applying
ethnomathematics
into teaching
and learning. Endogenous development
of teaching
may require reflection
of trivial
values and socio-mathematical
norms within the
school culture in each country and create new educational
practice. Endogenous development of
learning requires consideration
of what students are to acquire at the end of school education
where political,
economical
and cultural needs are intricately
intervened
with each other. All
these endogenous-ness are interrelated
to each other.
(2)
Which
points
to focus
The majority
of teachers
have sufficient
teaching
experience.
However, some of their
predictions
for student performance did not match the actual scores. This mismatch cast a
question over the assumption that teachers were aware of students'
background
and weakness.
The context and language of instruction
are still the points to focus for future researches. The
context was considered
extensively
in this research as follows.
-Teaching in school:
belief, and etc.
-Learning
curriculum,
environment
in school:
textbook,
supply
Family background:
parents' educational
in education, learning resources at home.
teacher's
of textbooks,
background,
competency,
teaching
Sociocultural
(3)
Future
and religious
factors:
beliefs
and learning
economic state, parents'
Learning environment in society:
public resources available
extra classes, private tutoring and self learning, and etc.
-Political
and economical
factors: national
market demands, examination, career ladder,
teaching
governance
and etc.
in education,
strategy,
resources,
belief
and etc.
and value
opportunities
and management strategies,
and values in society,
teacher's
for
labour
and etc.
research
Action research as a research method was proposed during the project. This is supported
by the
fact that this research involves many factors and their relation
is ever complicated.
So having
thick description
about present status and reorganizing
conjecture
is necessary. In any case,
flexibility
is required to adjust to the reality of each participating
country.
Reference
Berry, J.W. (1985)
'Learning
Mathematics
in a Second Language : some cross-cultural
issues',
For the Learning of Mathematics, 5(2), 1 8-23.
Fafunwa et al. (1989)
Education
in Mother Tongue: The Ife Primary Education Research
Project (1970-1978).
Nebres, B.F. (1988)
'School Mathematics
in the 1990s':
Recent Trends and the Challenge
to the
Developing
Countries',
Proceeding of the Sixth International
Congress on Mathematical
Education, 1 3-27.
UNESCO (2004) EFA Global Monitoring
Report 2005, Paris.
122
Post face
International
Cooperation
Project
Education
Final Report
Postface
Firstly I would like to congratulate
all those people involved in
have been directing
it, operating
it, participating
in it, supporting
it or
other way. A developmental
enterprise of this nature requires a great
of many people. This is always the case whatever the size and
endeavour.
this project - whether they
have been involved in any
deal of energy on the part
scale of the educational
This report is about the creation and execution ofa developmental
project in mathematics
education
involving
seven countries, each growing in its different
way. This is clearly
an
enormous undertaking,
and one can only admire the imagination
and organisational
abilities
of
those who have been responsible
for it. To engage in development
work in one country is
complex enough, but to do this with seven countries simultaneously
demands our admiration.
Even more than this is the highly praiseworthy
subscription
endogenous development, a process which has various significant
this educational
project, these would include the local determination
and local control over the development process. Also endogenous
values whilst exogenous development tends to ignore or subjugate
development is founded mainly on locally available
resources, and
and dynamise local resources which might otherwise be ignored or
of little value.
to the concept and ethic of
characteristics.
In relation to
of the development choices
development respects local
them. Finally,
endogenous
good schemes can revitalise
dismissed
by others as being
In this report we can read about some of the special local features and the questionnaire
and test results from the students in the different
countries. Whilst comparisons are made, and
similarities
and differences
noted, there is no attempt to produce 'league tables' of 'winners' and
'losers',
as happens in some international
comparisons.
In an endogenous project, by contrast, the aim is by comparing situations,
to identify
specific problem areas, and to search for strengths on which the local development can build.
By placing the research work in the hands of both local collaborators
and research students from
those countries, the project has sought to gain the best of both worlds. That is, the research
students are in touch with the latest ideas in both research methods and mathematics education,
while their local counterparts
are in continuous contact with the local situation.
Bringing
these
two together is the major strength and innovative part of this project.
project.
As a special advisor, I am very honoured to be able to contribute
in a small way to this
I wish success to all those involved in this promising
developmental
work.
Alan J. Bishop
Emeritus Professor
Centre for Science, Mathematics
and Technology
Faculty of Education
Monash University,
March 2007.
123
Education
Special
Contribution
International
Cooperation
Project
Special
1.
of Mathematics
Education
Teachers 'Final Report
Contribution
Numeracyand Mathematics
ACultural Perspective
on the Relationship
Alan J. Bishop
Monash University
1.1
Introduction
- numeracy
in a global
cultural
context
The concept of global citizenship
has gained currency in recent years. The mass
movementof people around the globe and the creation of regional entities such as the European
Union and ASEAN have generated significant
interest in the concept of global citizenship,
as
law-makers and curriculum policy advisors question the legitimacy
of traditional
notions of
citizenship.
These developments have tremendous implications
for those of us who work in
education, particularly
in mathematics education.
In Australia
for example, one must always be aware when considering
educational
that it is predominantly
a migrant country. A Report by the Public Affairs Section
Department of Immigration and Multicultural
Affairs, Canberra in 1996 states:
issues,
of the
"Today nearly one in four of Australia's
18.5 million
people was born overseas. In
1995-96 the number of settlers totalled
99 139. They came from more than 150 countries." In
the particular
cases of numeracy and lifelong education, I believe this recognition
is essential
for
any educational
initiatives
to succeed. Even in Japan, which is rarely considered
to be a
multi-cultural
society, the construct of 'culture' has strong meanings, and therefore important
implications
for numeracy education.
Whenever I am discussing
educational
issues I am always aware of three main constructs:
curriculum, teaching,
and learning. These are all of course embedded in social, cultural and
political
contexts which structure, support, and constrain them. When considering
numeracy
education, other people have tended to focus mainly on the learning aspects. So in this paper I
wish to focus on the other two main areas which I feel are often neglected,
and in particular on
these ideas:
1.culturally,
2.their
and socially
relation
- based numeracies,
to the mathematics
and
curriculum.
There are clearly issues and challenges
about defining and clarifying
the nature and role
of numeracy in today's society, and in particular
its relationship
with mathematics
(e.g.
FitzSimons
et al., 2003) Is numeracy a part of mathematics? Is it a simple form of mathematics?
Is it the same as mathematics? Is it radically
different from mathematics? I will offer my own
interpretation
of the relationship,
one which suggests some different ideas about a numeracy
curriculum than those normally given, and one that recognises
firstly
that there are many
numeracies in our societies,
and secondly that numeracies are socially and culturally
based.
1.2
The challenge
of culturally-based
mathematical
knowledge.
In the last 20 years, two main strands of research have developed
in mathematics
education.
The first was research on ethnomathematics,
which has fundamentally
changed
many of our ideas and constructs in considering
mathematics education (D'Ambrosio,
1985;
Gerdes, 1995). The most significant
influences have been in relation to:
125
International
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Project
of Mathematics
Education
Cultural
roots. Ethnomathematics
research has made us aware of the cultural starting
points, and different
histories,
of mathematics.
We now realise that Western Mathematics is
but one form among many, and that Wasan was a special form of mathematics in ancient Japan.
Interactions
between mathematics
and languages.
Ethnomathematics
research has
shown us that languages, and other symbol systems act as the principal
carriers of mathematical
ideas and values in different
cultures and societies.
Human interactions.
Ethnomathematics
research focuses on mathematical
activities
and
practices in society, and it thereby draws attention to the roles which people other than teachers
and learners play in mathematics education.
Values and
mathematical
activity
beliefs.
involves
Ethnomathematics
research
has made
values, beliefs and personal choices.
us realise
that
any
However the curriculum structures for mathematics
teaching that we generally see in
schools are typically
culture and value free, and have evolved to suit the preparation
of an elite
minority of students who will study mathematics
at university.
When considering
general
education, and especially
for those students who will never study mathematics at university,
this
elitist
curriculum
is highly
inappropriate.
It has contributed
significantly
to the widespread
problems of alienation
felt by many students, of all ages, towards mathematics.
But the problem is notjust
one of alienation,
it is much more to do with the fact that a
majority of the population
has been debarred from any discussion
or critique of the underlying
mathematical
constructs used in today's highly mathematically
formatted and structured society
(Skovsmose,
1994). Research therefore still needs to explore ways of making mathematics
curricula
more culturally
and socially
responsive,
in order to encourage more societal
participation
at the higher levels particularly
amongst cultural
and social minority
groups. I
believe that a numeracy-based curriculum is one way to emphasise the societal
and cultural
aspects of mathematics.
The second trend in research which has developed
in the past two decades, and which
relates to this discussion,
has demonstrated
the significance
for teaching,
learning,
and the
curriculum, of the conceptual idea of context, that is the context within which the mathematical
practice is situated.
Indeed the construct of "situated
mathematical
practice" is now attracting
a great deal of attention
(see for example, Kirshner & Whitson, 1997; Zevenbergen, 2000),
Much of this research work is based on the challenge
coming from studies of mathematical
activities
occurring outside school (e.g. Abreu et al., 2002; Lave & Wenger, 1991) where the
characteristics
of the practice are markedly different
from those in school.
These practices,
many of which have been the subject of ethnomathematics
research,
are to my mind the roots of numeracy education. This idea demonstrates powerfully
that the
context of school is markedly
different
from the contexts outside school where numeracy
practices occur. If we wish to develop numeracy curricula, how can we ensure that the school
context doesn't ignore, or indeed over-ride, the outside-school
numeracy contexts?
Bearing these two research trends in mind, the issue for researchers becomes: how can we
develop a numeracy-based mathematics curriculum for schools, one which satisfies
both the
needs of a modern democratic society for having fully educated and politically
contributing
adults, and the needs of the mathematical
research establishment
which still wants competent,
and in some cases brilliant,
mathematicians?
126
International
1.3
Cooperation
Numeracies
Project
and mathematical
of Mathematics
Education
Teachers ' Final Report
theory
Numeracy is being defined
in many different
ways, particularly
in its relation
to
mathematics, and particularly
in the context of adult education (see for example, Bessot and
Ridgway, 2000, and FitzSimons,
2002). There are those who define it as a part of mathematics,
often as simple or basic arithmetic.
Others see numeracy as far more than just arithmetic.
Still
others prefer to link numeracy to literacy rather than to mathematics (as UNESCO does), or to
consider numeracy as mathematical
literacy (Jablonka,
2003). A recent book from the USA
focuses on Quantitative
Literacy (Steen, 2001) although
in the text the word 'numeracy' is often
used in its place.
However I like to approach
the issue differently.
conceptualisation
of numeracy which meets three criteria:
æf it recognises
the existence
æf it clarifies
the links
æf it clarifies
the educational
Firstly
I believe
of many numeracies (Buckingham
between numeracies
and mathematics,
task facing those responsible
1997,
we need a
Tout, 2001)
and
for numeracy education.
The first aspect of the relationship
is to recognise the way that ethnomathematical
ideas
can be structured to enable them to be seen in relation to numeracy and to mathematics.
In my
original
research (Bishop,
1988)
I conceptualised
a structure based on six fundamental and
universal
activities,
which are found in all societies.
These activities
are the bases of the
numeracies in the societies
in which they appear. However they are also the bases of Western
Mathematics, and this is the heart of the relationship
we are seeking. Here are the six activities:
æf Counting:
This is the activity
concerned with the question "How many?" in all its
forms and variants eg. there are many ways of counting and of doing numerical calculations.
The mathematical
ideas derived from this activity
are numbers, calculation
methods, number
systems, number patterns, numerical methods, statistics,
etc.
æf Locating:
This activity concerns finding your way in the structured spatial world of
today, with navigating,
orienting
oneself and other objects, and with describing
where things are
in relation to one another. We use various forms of description
including
maps, figures, charts,
diagrams and coordinate
systems. This area of activity
is the 'geographical'
aspect of
mathematics. The mathematical
topics derived from this activity
are: dimensions,
Cartesian and
polar coordinates,
axes, networks, loci, etc.
æf Measuring:
"Howmuch?' is a question asked and answered in every society, whether
the amounts which are valued are cloth, food, land, money or time. The techniques
of measuring,
with all the different
units involved,
become more complex as the society
increases
in
complexity.
Mathematical
topics
derived
here are: order, size, units, measure systems,
conversion of units, accuracy, continuous quantities,
etc.
æf Designing:
Shapes are very important in the study of geometry and they appear to
derive from designing
objects to serve different
purposes. Here we are particularly
interested
in
how different shapes are constructed,
in analysing their various properties, and in investigating
the ways they relate to each other. The mathematical
topics derived are: shapes, regularity,
congruence, similarity,
drawing constructions,
geometrical
properties, etc.
æf Playing:
Everyone enjoys playing and most people take playing very seriously. Not all
play is important from a mathematical
viewpoint, but puzzles, logical paradoxes, some games,
and gambling
all involve
the mathematical
nature of many activities
in this category.
Mathematical
ideas derived here are: rules, procedures,
plans, strategies,
models, game theory,
etc.
127
International
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ofMathematics
Education
Teachers 'Final
Report
•E Explaining:
Trying to explain to oneself and to others why things happen the way they
do is a universal human activity. In mathematics we are interested in, for example, why number
calculations
work, and in which situations,
why certain geometric shapes do or do not fit
together, why one algebraic
result leads to another, and with different ways of symbolising
these
relationships.
Mathematical
topics derived here are: logic rules, proof, graphs, equations etc.
The next aspect of the relationship
appears when we realise that the cultural history of
Western Mathematics, which I shall continue to refer to by Mathematics with a capital 'M', (see
Bishop,
1988)
shows us that its essence
is its generality.
On the other hand the
ethnomathematics
literature
indicates
that numeracies are all about particular
practices.
To use
the word 'practices'
here is not just to refer to skills or algorithms,
nor to minimise numeracy as
just number work, but as with ethnomathematics,
and the six activities
above, numeracies
include meanings, and conceptualisations
also. No power relationship
between Mathematics and
numeracy is intended,
as both are powerful forms of knowledge in their own contexts. It is just
that the 'situated
context' of Mathematics is the abstract world of the theorist,
whereas the
'situated
context' of numeracy is the pragmatic world of the ethnomathematical
practitioner.
The relationship
between them is therefore best understood to my mind as one between
theory and practices, with numeracies being the practices and Mathematics being the theory
behind the practices.
Mathematics explains, theorises,
and clarifies
the rationales
underlying
those practices,
and also gets applied through various developments
of those practices.
This
conceptualisation
of the relationship
between the two forms of knowledge helps to construct a
meaning for numeracy, which meets the three criteria above.
As we can see with the activity
of 'explaining'
above, the question "Why does any
particular
numeracy practice work?" is an important key to understanding
the relationship,
and
in my view is the key to clarifying
the educational
task. Learners of all ages can bring many
numeracy, and ethnomathematical,
practices to their education,
mainly from their families
but
also from their wider society contacts - friends, media, other adults etc. However the naive
learner generally
lacks any understanding
of the underlying
Mathematical
theories that help to
explain those practices.
Without the Mathematical
theories they lack the tools to understand,
analyse and critique those practices.
Without these tools they become trapped by those practices,
as most adults currently are, without the understanding
to question and develop alternatives.
There are many 'why' questions that can be asked of school
current mathematics curriculum, such as these simple ones:
•E Why do the many different
succeed? It is because of the underlying
students
even within
practices
for counting,
addition,
subtraction
mathematical
structures of those algorithms.
•E Why do two negative numbers when multiplied
because of the wish to keep the rules of number theory
and negative integers.
together
consistent
give a positive
when dealing
the
etc. all
number? It is
with positive
•E Why can you divide one fraction
by another by 'turning
it upside
down and
multiplying'?
Because multiplying
the numerator and denominator of a fraction by the same
number doesn't change the value of that fraction.
•E Why does a three-legged
stool always balance, while a four-legged
one doesn't?
Because 3 points determine a plane.
Of course it may seem to many that asking these 'why' questions
is inappropriate
when
talking about the Mathematics curriculum, especially
since the demise of proof from the school
curriculum in many countries. Proofs are the ultimate mathematical
explanation.
But it is not
necessary to learn proofs of theorems just for regurgitation
at examination time. It is far more
important to understand the role of the process of 'proving'
in the development and education of
128
International
Cooperation
Project
of Mathematics
the numerate and Mathematically
literate student. 'Why' questions
Mathematics
and numeracy practice,
and within Mathematics
questions will often be found by proving.
Education
can be asked at any level of
itself the answers to those
Thus my conceptualisation
of the relationship
between numeracies and Mathematics is
that those numeracies are culturally,
socially,
and historically
determined,
and practically
powerful. Mathematics on the other hand has developed culturally
in a different
way, and in a
different
context, as an extremely powerful way of theorising,
explaining,
and extending our
knowledge of the world.
The implications
main ones:
for school
mathematics
curricula
are many, but here are some of the
•E These curricula
need to recognise,
and to be built around, the
practices which the learners have learnt at home and in the wider society, and
to the education situation.
Thus not all the curricular content can be determined
must be determined
locally within a national or regional framework. The six
can provide such a framework.
many numeracy
which they bring
centrally.
Much
activities
model
•E These curricula need also to introduce learners to the many numeracy practices which
powerful others in society practise and which can impinge on their lives in crucial ways. There
are of course many of these in adult life, but the criterion
should be whether, and how, they
impinge on the lives of the learners.
•E These numeracy practices need to be clarified,
theorised,
and critiqued
in terms of the
Mathematical
theories
behind them. This is to reverse the usual approach of teaching
the
Mathematical
theory followed by its applications.
The particular
curricular challenge which this
implication
presents, is how to create sensible
sequential
designs which can help to structure
that approach. This is a challenge
ofa high order.
•E Generally,
mathematical
ideas should always
numeracy practices where they are used. The complaint
meaningless
and abstract subject, which reflects the need
between the two were better understood by teachers, and
not find so much alienation
in our schools.
be related to the various contexts and
is often voiced that mathematics
is a
for this implication.
If the relationship
then by the learners, perhaps we would
References
Abreu,
G. de, Bishop,
A.J. & Presmeg, N.C. (2002)
mathematics learning. Dordrecht, Holland : Kluwer.
Transitions
D'Ambrosio,
U. (1985)
Ethnomathematics
and its place in the
mathematics. For the learning of mathematics, 5( 1 ), 44-48
Bessot,
A. & Ridgway,
Holland:
Kluwer
J. (Eds.)
(2000)
Bishop,
A.J. (1988)
Mathematical
education. Dordrecht,
Holland:
Education
for mathematics
enculturation:
Kluwer
a cultural
between
history
contexts
for
and pedagogy
of
in the workplace.
perspective
Dordrecht,
on mathematics
Buckingham, E.A. (1997)
Specific
and generic numeracies of the workplace:
How is numeracy
learnt and used by workers in production
industries,
and what learning/working
environments
promote this?
Burwood,
Victoria,
Australia:Centre
for Studies
in
Mathematics,
Science, and Environmental
Education, Deakin University
FitzSimons,
G. (2002)
What counts
vocational
FitzSimons,
as mathematics:
Holland: Kluwer
G, Coben, D. & O'Donoghue,
J. (Eds.)
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Technologies
(2003)
Lifelong
of power in adult
mathematics
learning.
and
In
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Cooperation
A.J.Bishop,
international
Kluwer
Gerdes, P. ( 1 995)
Project
M.A.Clements,
C.Keitel,
handbook
of mathematics
Ethnomathematics
J.Kilpatrick,
education
and education
of Mathematics
& F.K.S.Leung,
(Eds)
(pp. 103-142)
Dordrecht,
in Africa, Sweden:
Jablonka, E. (2003)
Mathematical
literacy.
In A.J.Bishop,
& F.K.S.Leung,
(Eds)
Second international
(pp. 75-102) Dordrecht, Holland:
Kluwer
J., & Wenger, E. (1991).
Situated
learning:
Cambridge:
Cambridge
University
Press.
Stockholm
Second
Holland:
University.
M.A.Clements, C.Keitel,
J.Kilpatrick,
handbook
of mathematics
education
Kirshner,
D. & Whitson, J.A.(1997)
Situated
cognition:
social,
perspectives.
New Jersey: Lawrence Erlbaum Associates
Lave,
Education
Legitimate
semiotic,
peripheral
and psychological
participation.
Skovsmose, O. (1994)
Towards a philosophy
of critical
mathematics
education.
Holland:
Kluwer
Steen, L.A. (ed.) (2001)
Mathematics
and democracy:
the case for quantitative
Washington:
National
Council on Education and the Disciplines.
Dordrecht,
literacy.
Tout, D. (2001)
What is numeracy? What is mathematics?
In GE.FitzSimons,
J.O'Donaghue
&
D.Coben (Eds.)
Adult and lifelong
education
in mathematics.
Melbourne,
Australia:
Language Austral ia
Zevenbergen,
R. (2000)
Ethnography
and the situatedness
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209-224)
Dordrecht,
Holland:
Kluwer
130
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2
Cooperation
The World
Project
Role of Culture
in Mathematics
Education
Teachers 'Final
Report
Education1
NormaPresmeg
Illinois
State University
Mathematics education has experienced
a major revolution in perceptions
(cf. Kuhn,
1970) comparable to the Copernican revolution that no longer placed the earth at the center of
the universe. This change has implicated
beliefs
about the role of culture in the historical
development of mathematics
(Eves, 1990), in the practices
of mathematicians
(Civil,
2002;
Sfard, 1997), in its political
aspects (Powell & Frankenstein,
1997), and hence necessarily
in its
teaching
and learning
(Bishop,
1988a;
Bishop & Abreu, 1991; Bishop
& Pompeu, 1991;
Nickson, 1992). The change has also influenced
methodologies
that are used in mathematics
education research (Pinxten,
1994). Researchers now increasingly
concede that mathematics,
long considered
value- and culture-free,
is indeed a cultural product, and hence that the role of
culture-with
all its complexities
and contestations-is
an important aspect of mathematics
education.
Topics that are central in addressing
this role of culture are those arising from and
extending
the notions of ethnomathematics
and everyday cognition
(Nunes, 1992, 1993).
Various broad theoretical
fields are relevant. Some of the theoretical
notions that are apposite
are rooted in-but not confined to-situated
cognition
(Kirshner
& Whitson, 1997; Lave &
Wenger, 1991; Watson, 1998), cultural
models (Holland
& Quinn, 1987), notions of cultural
capital
(Bourdieu,
1995),
didactical
phenomenology
(Freudenthal,
1973, 1983),
and Peircean
semiotics (Peirce,
1992, 1998). From this partial list, the breadth of this developing
field may be
recognized.
This paper does not attempt to treat the general theories in detail:
The interested
reader is referred to the original
authors. Instead, some key notions that have explanatory power
or usefulness are the central focus, which is the role of culture in learning and teaching
mathematics all over the world. The seminal work of Ascher (1991,
2002),
Bishop (1988a,
1988b),
D'Ambrosio
(1985,
1990)
and Gerdes
(1986,
1988a,
1988b,
1998)
on
ethnomathematics
is still centrally
relevant and thus is treated in some detail.
In the last decade, the field of research into the role of culture in mathematics education
has evolved from "ethnomathematics
and everyday cognition"
(Nunes, 1992), although
both
ethnomathematics
and everyday cognition
are still important
topics of investigation.
The
developments
have rather consisted
in a broadening
of the field, clarification
and evolution of
definitions,
recognition
of the complexity
of the constructs and issues, and inclusion
of social,
critical,
and political
dimensions
as well as those from cultural
psychology
involving
valorization,
identity,
and agency (Abreu, Bishop, & Presmeg, 2002). This paper in its scope
cannot do full justice
to political
and critical views of mathematics education (see Mellin-Olsen,
1987;
Skovsmose, 1994; Vithal, 2003, for a full treatment),
but some of the landscapes from
these fields-as Vithal calls them-are used to deepen and problematize
aspects of the treatment
of culture in mathematics education.
This paper has four sections.
The first addresses
an introduction
to issues and
definitions
of key notions involving
culture in mathematics
education. The organization
of the
second and third sections uses the research framework of Brown and Dowling (1998).
In this
framework, resonating theoretical
and empirical
fields surround and enclose the central research
topic, and the description
involves layers of increasing
specificity
as it zooms in on details of
This paper is an abridged
version ofPresmeg's
Chapter 1 1, "The role of culture in teaching
and
learning mathematics",
in F. K Lester (Ed.), Second handbook of research on mathematics teaching
learning, pp. 435-458.
Charlotte,
North Carolina:
Information
Age Publishing,
2007.
131
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Cooperation
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of Mathematics
Education
Teachers ' Final
Report
the problematic
and problems of the research issues, the empirical
settings,
and the results of the
studies,
only to zoom out again at the end in order to survey the issues in a broader field,
informed by the results of the research studies examined, in order to see where further research
on culture might be headed. Thus section 2 addresses theoretical
fields that incorporate
culture
specifically
in mathematics education;
section 3 addresses salient empirical
fields, settings,
and
some details of results of research on culture in mathematics education, and their implications
for the teaching
and learning of mathematics. Using a broader perspective,
the fourth (final)
section collects
and elaborates suggestions
for possible
directions
for future research on culture
in mathematics education.
2.1
Definitions
and Significance
of Culture
in Mathematics
Education
Why is it important to address definitions
carefully in considering
the role of culture in
teaching and learning
mathematics?
As in other areas of reported research in mathematics
education, different
authors use terminology
in different
ways. Particularly
notions such as
culture, ethnomathematics,
and everyday mathematics have been controversial;
they have been
contested and given varied and sometimes competing interpretations
in the literature.
Thus these
constructs must be problematized,
not only for the sake of clarification,
but more importantly
for the roles they have played, and for their further potential
to be major focal points in
mathematics education research and practices. Further, especially
in attempts to bridge the gap
between the formal mathematics taught in classrooms and that used out-of-school
in various
cultural practices, what counts as mathematics assumes central importance (Civil,
1995, 2002;
Presmeg, 1998a).
Thus ontological
aspects of the nature of mathematics
itself must also be
addressed.
(1 )
Culture
In 1988, when Bishop published
his book, Mathematical
Enculturation:
A Cultural
Perspective on Mathematics Education (1988a)
and an article that summarized these ideas in
Educational Studies in Mathematics (1988b,
now reprinted as a classic in Carpenter, Dossey, &
Koehler, 2004), the prevailing
view of mathematics was that it was the one subject in the school
curriculum
that was value- and culture-free,
notwithstanding
a few research studies
that
suggested the contrary (e.g., Gay & Cole, 1967; Zaslavsky,
1973). Along with those of a few
other authors (notably,
D'Ambrosio),
Bishop's
ideas have been seminal in the recognition
that
culture plays a pivotal role in the teaching
and learning of mathematics,
and his insights
are
introduced repeatedly
in this paper.
Whole books have been written about definitions
of culture (Kroeber & Kluckhohn,
1952).
Grappling
with the ubiquity
yet elusiveness
of culture, Lerman (1994)
confronted the
need for a definition
but could not find one that was entirely satisfactory.
Yet, as he pointed out,
culture is "ordinary.
It is something
that we all possess and that possesses us" (p. 1). Bishop
(1988a)
favored the following
definition
of culture in analyzing
its role in mathematics
education:
Culture consists of a complex of shared understandings
which serves as a medium through
which individual
human minds interact
in communication
with one another, (p. 5,
citing Stenhouse, 1967, p. 16)
This definition
highlights
the communicative function of culture that is particularly
relevant in
teaching and learning. However, it does not focus on the continuous renewal of culture, the
dynamic aspect that results in cultural
change over time. This dynamic aspect caused Taylor
(1996)
to choose the "potentially
transformative
view" (p. 151) of cultural
anthropologist
Clifford
Geertz (1973),
for whom culture consists of "webs of significance"
(p. 5) that we
ourselves have spun.
This potentially
transformative
view assumes particular
importance in the light of the
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necessity for negotiating
social norms and sociomathematical
norms in mathematics classrooms
(Cobb & Yackel, 1996). The culture of the mathematics classroom, which was brought to our
attention
as being significant
(Nickson,
1994), is not monolithic
or static but continuously
evolving,
and different
in different
classrooms
as these norms become negotiated.
The
mathematics
classroom itself is one arena in which culture is contested,
negotiated,
and
manifested
(Vithal,
2003),
but there are various levels of scale. Bishop's
(1988a)
view of
mathematics education as a social process resonates with a transformative,
dynamic notion of
culture. He suggested that five significant
levels of scale are involved in the social aspects of
mathematics education. These are the cultural, the societal,
the institutional,
the pedagogical,
and the individual
aspects (1988a,
p. 14). Culture here is viewed as an all-encompassing
umbrella construct that enters into all the activities
ofhumans in their communicative and social
enterprises.
In addition
to a view of culture in this macroscopic aspect, as in these levels of scale,
culture as webs of significance
may be central also in the societal,
institutional,
and pedagogical
aspects of mathematics education considered as a social process. Thus researchers may speak of
the culture of a society, ofa school, or of an actual classroom. Culture in all of these levels of
scale impinges on the mathematical learning of individual
students.
Notwithstanding
these general definitions
of culture,
the word with its various
characterizations
does not have meaning in itself. Vithal and Skovsmose (1997)
illustrated
this
point starkly
by pointing
out that interpretations
of culture (and by implication
also
anthropology)
were used in South Africa to justify the practices of apartheid.
They extended the
negative connotations
still attached to this word in that context to suggest that ethnomathematics
is also suspect (as suggested
in the title of their article: "The End of Innocence: A Critique of
'Ethnomathematics'").
Some aspects of their critique are taken up in later sections.
1 994),
Howare culture and society interrelated?
The words are not interchangeable
although
connections exist between these constructs:
(Lerman,
One would perhaps think of gender stereotypes
as cultural,
but of 'gender'
as socially
constructed.
One would talk of the culture
of the community
of
mathematicians,
treating
it as monolithic
for a moment, but one would
also talk, for example, of the social outcomes of being a member of that
group, (p. 2)
Alan Bishop stated succinctly,
on several occasions (personal
communication, e.g., July, 1985),
that society involves various groups ofpeople, and culture is the glue that binds them together.
This informal characterization
resonates with Stenhouse's definition
of culture as a complex of
shared understandings,
and also with that of Geertz, as webs of significance
that we ourselves
have spun. Considering
the cultures of mathematics classrooms, Nickson (1994)
wrote of the
"invisible
and apparently
shared meanings that teachers and pupils bring to the mathematics
classroom and that govern their interaction
in it" (p. 8). Values, beliefs,
and meanings are
implicated
in these "shared invisibles"
(p. 18) in the classroom. Nickson saw socialization
as a
universal process, and culture as the content of the socialization
process, which differs from one
society to another, and indeed, from one classroom to another.
Part of culture as webs of significance,
of what counts as mathematics.
(2)
taken at various levels,
are the prevailing
notions
Mathematics
As Nickson (1994)
pointed out, "one of the major shifts in thinking
in relation
to the
teaching and learning of mathematics in recent years has been with respect to the adoption of
differing
views of the nature of mathematics as a discipline"
(p. 10). Nickson characterized
this
cultural shift as moving from aformalist tradition
in which mathematics is absolute-consisting
of "immutable truths and unquestionable
certainty" (p. ll)-without
a human face, to one of
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growth and change, under persuasive
influences
such as Lakatos's
(1976)
argument that
"objective
knowledge"
is subject
to proofs and refutations
and thus that mathematical
knowledge has a strong social component. That this shift is complex and that both views of
mathematics are held simultaneously
by many mathematicians
was argued by Davis and Hersh
(1981).
The formalist
and socially
mediated
views of mathematics
resonate with the two
categories
of absolutist
and fallibilist
conceptions
discussed
by Ernest (1991).
Contributing
the
notion of mathematics as problem solving, the Platonist,
the problem-solving,
and the fallibilist
conceptions
are categories
reminiscent
of the teachers' conceptions
of mathematics that Alba
Thompson, already
in 1984,
gave evidence
were related
to instructional
practices
in
mathematics classrooms.
Both Civil (1990,
1995, 2002; Civil & Andrade, 2002) in her "Funds of Knowledge"
project
and in her later research with colleagues
into ways of linking
home and school
mathematical
practices,
and Presmeg (1998a,
1 998b, 2002b)
in her use of ethnomathematics
in
teacher education and research into semiotic chaining as a means of building
bridges between
cultural
practices
and the teaching
and learning
of mathematics
in school,
described
the
necessity of broadening
conceptions
of the nature of mathematics
in these endeavors. Without
such broadened definitions,
high-school
and university
students alike are naturally
inclined
to
characterize
mathematics according
to what they have experienced
in learning
institutional
mathematics-more often than not as "a bunch of numbers" (Presmeg, 2002b). On the one hand,
such limited views of the nature of mathematics inhibit the recognition
of mathematical
ideas in
out-of-school practices.
On the other hand, if definitions
of what counts as mathematics are too
broad, then the "everything
is mathematics"
notion may trivialize
mathematics
itself, rendering
the definition
useless. In examining the mathematical
practices of a group of carpenters in Cape
Town, South Africa, Millroy (1992)
expressed this tension well, as follows:
[It] became clear to me that in order to proceed with the exploration
of the mathematics
of
an unfamiliar
culture, I would have to navigate a passage between two dangerous areas. The
foundering
point on the left represents
the overwhelming
notion that 'everything
is
mathematics'
(like being swept away by a tidal wave!) while the foundering
point on the
right represents the constricting
notion that 'formal academic mathematics is the only valid
representation
of people's mathematical
ideas' (like being stranded on a desert island!).
Part
of the way in which to ensure a safe passage seemed to be to openly acknowledge that when
I examined the mathematizing
engaged in by the carpenters there would be examples of
mathematical
ideas and practices that I would recognize and that I would be able to describe
in terms of the vocabulary of conventional
Western mathematics. However, it was likely that
there would also be mathematics that I could not recognize and for which I would have no
familiar descriptive
words, (pp. 1 1-13)
Some definitions
of mathematics
that achieve a balance between these two extremes are as
follows. Mathematics is "the language and science of patterns" (Steen, 1990, p. i i). Steen's
definition
has been taken up widely in reform literature
in the USA (National
Council
of
Teachers of Mathematics
[NCTM], 1989, 2000).
Opening the gate to recognizing
a human
origin of mathematics,
Saunders MacLane called mathematics
"the study of formal abstract
structures arising from human experience" (as cited in Lakoff, 1987, p. 361). According to Ada
Lovelace, mathematics
is the systematization
of relationships
(as described
by Noss, 1997). All
of these definitions
strike some sort of balance between the human face of mathematics and its
formal aspects. Going beyond Steen's well-known pattern definition
in the direction
of stressing
abstraction,
in a critique
of ethnomathematics,
Thomas (1996)
defined mathematics
as "the
science of detachable
relational
insights"
(p. 17). He suggested a useful distinction
between real
mathematics (as characterized
in his definition),
and proto-mathematics
(the category in which
he placed
ethnomathematics).
In the next section,
Barton's (1996)
characterization
of
ethnomathematics,
which resolves many of these issues and clarifies
this dualism,
is presented
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along with some evolving definitions
consult Barton's original
article.)
(3)
ofMathematics
of ethnomathematics.
(For
Education
details,
the reader
should
Ethnomathematics
What is ethnomathematics'}
In his illuminating
article,
Barton ( 1 996)
wrote as follows:
In the last decade, there has been a growing literature
dealing with the relationship
between
culture and mathematics,
and describing
examples of mathematics
in cultural
contexts.
What is not so well-recognised
is the level at which contradictions
exist within this
literature:
contradictions
about the meaning of the term 'ethnomathematics'
in particular,
and also about its relationship
to mathematics as an international
discipline,
(p. 201)
Barton pointed out that difficulties
in defining
ethnomathematics
relate to three categories:
epistemological
confusion, "problems with the meanings of words used to explain ideas about
culture and mathematics" (p. 201); philosophical
confusion, the extent to which mathematics is
regarded
as universal;
and confusion
about the nature of mathematics.
The nature of
mathematics is part of its ontology, and because both ontology and epistemology
are branches
of philosophy,
all of these categories may be regarded as philosophical
difficulties.
The strength
of Barton's resolution
of the difficulties
lies in his creation of a preliminary
framework (he
admitted that it might need revision)
whereby the differing
views can be seen in relation to each
other. His triadic framework is an "Intentional
Map" (p. 204) with the three broad headings of
mathematics, mathematics education, and society (cf. the whole day of sessions dedicated
to
these broad areas at the 6th International
Education held in Budapest,
Hungary, in 1988). The seminal writers whose definitions
of ethnomathematics
he considered in
detail and placed in relation to this framework were Ubiratan D'Ambrosio in Brazil, Paulus
Gerdes in Mozambique, and Marcia Ascher in the USA.
As Barton (1996)
pointed
out, D'Ambrosio's
prolific
writings
on the subject
of
ethnomathematics
have influenced
the majority of writers in this area. Thus on the Intentional
Map, although D'Ambrosio's
work (starting with his 1984 publication)
falls predominantly
in
the socio-anthropological
dimension
between society and mathematics, some aspects of his
concerns can be found in all of the dimensions.
In his later work, he increasingly
used his model
to analyze "the way in which mathematical
knowledge is colonized
and how it rationalizes
social divisions
within societies
and between societies"
(Barton, 1996, p. 205). In his early
writing,
D'Ambrosio (1984)
defined ethnomathematics
as the way different
cultural
groups
mathematize-count,
measure, relate, classify,
and infer. His definition
evolved over the years,
to include a changing form of knowledge manifest in practices
that change over time. In 1985,
he defined ethnomathematics
as "the mathematics which is practiced
among identifiable
cultural
groups" (p. 1 8). Later, in 1987, his definition
of ethnomathematics
was "the codification
which
allows a cultural group to describe, manage, and understand reality" (Barton, 1996, p. 207).
D'Ambrosio's
in full in the following
( 1 99 1) well-known
passage.
etymological
definition
of ethnomathematics
is given
The main ideas focus on the concept of ethnomathematics in the sense that follows. Let me
clarify at the beginning
that this term comes from an etymological
abuse. I use mathema(ta)
as the action of explaining
and understanding
in order to transcend and of managing and
coping with reality in order to survive. Man has developed throughout each one's own life
history
and throughout
the history
of mankind technics (or tics) of mathema in very
different
and diversified
cultural environments, i.e., in the diverse ethno's. So, in order to
satisfy the drive towards survival and transcendence
in diverse cultural environments, man
has developed and continuously
develops, in every new experience, ethno-mathema-tics.
These are communicated vertically and horizontally
in time, respectively
throughout
history
and through conviviality
and education, relying on memory and on sharing experiences and
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knowledges.
For the reasons of being more or less effective,
more or less powerful and
sometimes even for political
reasons, some of these different
tics have lasted and spread
(ex.: counting,
measuring)
while others have disappeared
or been confined to restricted
groups. This synthesizes
my approach to the history of ideas, (p. 3)
As in some of his other writings
(1985,
1987,
1990),
D'Ambrosio
is in this definition
characterizing
ethnomathematics
as a dynamic, evolving system of knowledge-the
"process of
knowledge-making"
(Barton, 1996, p. 208), as well as a research program that encompasses the
history of mathematics.
Returning to Barton's Intentional
Map, the work of Paulus Gerdes is "practical,
and
politically
explicit,"
concentrated
in the mathematics education area of the Map (Barton, 1996,
p. 205). Gerdes's definition
of ethnomathematics
evolved from the mathematics
implicit
or
"frozen" in the cultural practices
of Southern Africa (1986),
to that of a mathematical
movement
that involves research and anthropological
reconstruction
(1994).
The work of mathematician
Marcia Ascher (1991,
1995, 2002), while overlapping
with that of Gerdes to some extent, falls
closer to the mathematics area on the Map, concerned as it is with cultural mathematics.
Her
definition
is that ethnomathematics
is "the study and presentation
of the mathematical
ideas of
traditional
peoples" (1991, p. 1 88). When Ascher (1991)
worked out the kinship relations
of the
Warlpiri,
say, in mathematical
terms, she acknowledged
that she was using her familiar
"Western" mathematics. In that sense her ethnomathematics
is subjective:
The Warlpiri would
be unlikely
to view their kinship
system through her lenses. Referring
to mathematics
and
ethnomathematics,
she stated, "They are both important, but they are different.
And they are
linked"
(Ascher
& D'Ambrosio,
1994, p. 38). In this view, there is no need to view
ethnomathematics
as "proto-mathematics"
(Thomas, 1996), because it exists in its own right.
Finally,
Barton (1996)
found a useful metaphor to sum up the similarities
and
differences
between the views of ethnomathematics
held by these three proponents:
"For
D'Ambrosio
it is a window on knowledge itself;
for Gerdes it is a cultural
window on
mathematics;
and for Ascher it is the mathematical
window on other cultures" (p. 213). These
three windows are distinguished
by the standpoint
of the viewer, and by what is being viewed,
in each case. Although
not eliminating
the duality
of ethnomathematics
as opposed to
mathematics (of mathematicians),
these three distinct
windows represent approaches each of
which has something to offer. Taken together, they contribute
a broadened lens on the role of
culture in teaching and learning mathematics.
Several other writers in the field of ethnomathematics
have acknowledged
the need and
attempted to define ethnomathematics.
Scott (1985)
regarded ethnomathematics
as lying at the
confluence
of mathematics
and cultural
anthropology,
"mathematics
in the environment
or
community," or "the way that specific cultural groups go about the tasks of classifying,
ordering,
counting, and measuring" (p. 2). Several definitions
of ethnomathematics
highlight
some of the
"environmental
activities"
that Bishop (1988a)
viewed as universal,
and also "necessary and
sufficient for the development of mathematical
knowledge" (p. 1 82), namely counting, locating,
measuring, designing,
playing,
and explaining.
One further definition
brings back the problem,
hinted at in the foregoing account, of ownership of ethnomathematics.
Whose mathematics is it?
Ethnomathematics
refers to any form of cultural knowledge or social activity characteristic
of a social and/or cultural group, that can be recognized
by other groups such as 'Western'
anthropologists,
but not necessarily
by the group of origin, as mathematical
knowledge or
mathematical
activity. (Pompeu, 1994, p. 3)
This definition
resonates with Ascher's, without fully solving the problem of ownership. The
same problem appears in definitions
of everyday mathematics,
considered next.
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Project
ofMathematics
Education
Mathematics
Following
on from the description
of everyday cognition (Nunes, 1992, 1993) and
important early studies that examined the use of mathematics
in various practices,
such as
mathematical
cognition
of candy sellers in Brazil (Carraher, Carraher, & Schliemann,
1985;
Saxe, 1991), constructs and issues are still being questioned.
In this area, too, clarification
of
definitions
is being sought, along with deeper consideration
of the scope of the issues and their
potential
and significance
for the classroom learning of mathematics.
Brenner and Moschkovich
(2002)
raised
the following
questions.
What do we mean by everyday mathematics! How is everyday mathematics
related to
academic mathematics1? What particular
everyday practices
are being brought
into
mathematics
classrooms?
What impact do different
everyday practices actually
have in
classroom practices?"
(p. v)
In a similar vein, and with the benefit of2 decades of research experience in this area, Carraher
and Schliemann (2002) examined how their perceptions
had evolved, as they explored the topic
of their chapter, "Is Everyday Mathematics Truly Relevant to Mathematics Education?"
All the authors of chapters in the monograph edited by Brenner and Moschkovich
(2002)
in one way or another set out to explore these and related questions.
Several of these
authors pointed out that it is problematic
to oppose everyday and academic mathematics,
for
several reasons. For one thing, for mathematicians
academic mathematics
is an everyday
practice
(Civil,
2002; Moschkovich,
2002a). For another, studies of everyday mathematical
practices
in workplaces reveal a complex interplay with sociocultural
and technological
issues
(FitzSimons,
2002).
In the automobile
production
industry,
variations
in the mathematical
cognition
required of workers have less to do with the job itself than with the decisions
of
management concerning production
procedures and organization
of the workplace.
Highly
skilled
machinists
display spatial and geometric knowledge that goes beyond what is commonly
taught in school: In contrast, assembly-line
workers and some machine operators find few if any
mathematical
demands in their work, which is deliberately
stripped
of the need for decisions
involving knowledge of mathematics beyond elementary counting (Smith, 2002). The complex
relationship
between use of technology
and the demand for mathematical
thinking
in the
workplace is a theme that is explored in a later section.
Another aspect that is again apparent in all the chapters of Brenner and Moschkovich's
monograph is the importance ofperceptions
and beliefs about the nature of mathematics, both in
the microculture
of classroom practices (Brenner, 2002; Masingila,
2002) and in the broader
endeavor to bridge the gap between mathematical
thinking
in and out of school (Arcavi, 2002;
Civil, 2002; Moschkovich,
2002a). An essential element in all of these studies is the concern to
connect knowledge of mathematics in and out of school. (This issue is revisited
later.) Because
of the difficulties
surrounding
the construct everyday mathematics the terminology
that will be
adopted in this chapter follows Masingila
(2002),
who referred to in-school and out-of-school
mathematics practices (p. 3 8).
The developments described
in this section parallel
the genesis of the movement away
from purely psychological
cognitive
and behavioral
frameworks for research in mathematics
education, towards cultural frameworks that embrace sociology, anthropology,
and related fields,
including
political
and critical
perspectives.
The following
section introduces
some relevant
theoretical
issues and lenses that have been used to examine some of these developments.
2.2
Theoretical
Fields
That
Incorporate
Culture
in Mathematics
Education
The notion of theoretical
and empirical
fields is drawn from Brown and Dowling (1998)
and provides a useful framework for characterizing
components of research. This section
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addresses
some theoretical
fields pertinent
to culture
in the teaching
and learning
of
mathematics. Their instantiation
in empirical
studies is described
in the next section. The reader
is reminded again to consult the original
authors for a full treatment of theoretical
fields that are
introduced
in this section, which has as its purpose a wide but by no means exhaustive view of
the scope of theories that are available
for work in this area.
As suggested
in Barton's (1996)
sense-making
article introduced
in the previous
section, in the last 2 decades there has already been considerable
movement in theoretical
fields
regarding
the interplay
of culture and mathematics. One such movement is discernible
in the
definitions
of ethnomathematics
given by D'Ambrosio,
Gerdes, and Ascher, as their theoretical
formulations
moved from more static definitions
of ethnomathematics
as the mathematics
of
different
cultural
groups, to characterizations
of this field as an anthropological
research
program that embraces not only the history of mathematical
ideas of marginalized
populations,
but the history of mathematical
knowledge itself (see previous section).
D'Ambrosio (2000, p.
83) called this enterprise
historiography.
(1 )
Historiography
Moving beyond earlier theoretical
formulations
of ethnomathematics,
its importance as
a catalyst for further theoretical
developments
has been noted (Barton,
1996). D'Ambrosio
played a large role and served as advisory editor in the enterprise that resulted in Helaine Selin's
(2000)
edited book, Mathematics
Across Cultures:
The History ofNon-Western Mathematics.
The chapters in this book are global in scope and record the mathematical
thinking
of cultures
ranging from those of Iraq, Egypt, and other predominantly
Islamic
countries;
through the
Hebrew mathematical
tradition;
to that of the Incas, the Sioux of North America, Pacific
cultures, Australian
Aborigines,
mathematical
traditions
of Central and Southern Africa; and
those of Asia as represented by India, China, Japan, and Korea. As can be gleaned from the
scope of this work, D'Ambrosio's original concern to valorize the mathematics of colonized and
marginalized
people (cf. Paolo Freire's
Pedagogy of the Oppressed in 1970/1997)
has
broadened to encompass a movement that is both archeological
and historical
in nature, based
on the theoretical
field of "historiography"
and visions
of world knowledge through the
"sociology
of mathematics" (D'Ambrosio,
2000, pp. 85-87).
Although
these antedated
D'Ambrosio's
program, earlier
studies
such as Claudia
Zaslavsky's
(1973)
report on the counting
systems of Africa and Glendon Lean's (1986)
categorization
of those ofPapua New Guinea (see also Lancy, 1983), could also be thought of as
historiography,
as could anthropological
research such as that of Pinxten, van Dooren, and
Harvey (1983)
who documented Navajo conceptions of space. Also in the cultural anthropology
tradition,
Crump's (1994)
research on the anthropology
of numbers is another fascinating
example of historiography.
More recent studies
such as some of those collected
as
Ethnomathematics
in the book by Powell and Frankenstein
(1997)
and the work of Marcia
Ascher (1991,
1995, 2002)
also fall into this category. Many of the cultural anthropological
studies of various mathematical
aspects of African practices,
such as work on African fractals
(Eglash,
1999);
lusona of Africa (Gerdes,
1997);
and women, art, and geometry of Africa
(Gerdes, 1998), may also be regarded as historiography.
As part of ethnomathematics
conceived
as a research program, this ambitious
undertaking of historiography
is designed
to address
lacunae in the literature
on the history of mathematical
thought through the ages. Much of the
work of members of the International
Study Group on Ethnomathematics
(founded
in 1985),
and of the North American Study Group on Ethnomathematics
(founded in 2003)-including
research by Lawrence Shirley, Daniel Orey, and many others-could
be placed in the category
ofhistoriography
(see the list of some of the available
web sites following
the references).
Because historiography
been ignored or undervalued
addresses some of the world's mathematical
systems that have
in mathematics
classrooms, it reminds us that there is cultural
138
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Cooperation
capital involved
relevant theories
(2)
Cultural
Project
in the power relations
associated
with access
are outlined in the following paragraphs.
Capital
and
in school
Education
Teachers ' Final
Report
mathematics.
Some
Habitus
The usefulness of the theoretical
field outlined by Bourdieu (1995)
for issues of culture
in the learning of mathematics has been indicated
in research on learner's transitions
between
mathematical
contexts (Presmeg, 2002a).
Resonating
with D'Ambrosio's
original
issues of
concern (although
D'Ambrosio did not use this framework), Bourdieu's
work belongs in the
field of sociology.
The relevance of this work consists in "the innumerable and subtle strategies
by which words can be used as instruments
of coercion and constraint,
as tools of intimidation
and abuse, as signs of politeness,
condescension,
and contempt" (Bourdieu,
1995, p. 1, editor's
introduction).
This theoretical
field serves as a useful lens in examining empirical
issues related
to the social inequalities
and dilemmas faced by mathematics
learners as they move between
different
cultural contexts, for example in the transitions
experienced
by immigrant children
learning mathematics in new cultural settings (Presmeg, 2002a). This field embraces Bourdieu's
notions of cultural capital, linguistic
capital, and habitus. Bourdieu (1995)
used the ancient
Aristotelian
term habitus to refer to "a set of dispositions
which incline agents to act and react in
certain ways" (p.12).
Such dispositions
are part of culture viewed as a set of shared
understandings.
Various forms of capital are economic capital (material wealth), cultural capital
(knowledge,
skills,
and other cultural acquisitions,
as exemplified
by educational
or technical
qualifications),
and symbolic capital (accumulated
prestige
or honor) (p. 1 4). Linguistic
capital
is not only the capacity to produce expressions that are appropriate
in a certain social context,
but it is also the expression
of the "correct" accent, grammar, and vocabulary. The symbolic
power associated
with possession of cultural, symbolic, and linguistic
capital has a counterpart
in the symbolic
violence experienced
by individuals
whose cultural
capital
is devalued.
Symbolic violence is a sociological
construct. In that capacity it is a powerful lens with which to
examine actions of a group and ways in which certain types of knowledge are included
or
excluded in what the group counts as knowledge (for examples embracing the learning of
mathematics, see the chapters in Abreu, Bishop, & Presmeg, 2002).
(3)
Borderland
Discourses
The notion of symbolic violence leaves a possible theoretical
gap relating to the ways in
which
individuals
choose
to construct,
or choose
not to construct,
particular
knowledge-mathematical
or otherwise.
Bishop (2002a)
gave examples both of immigrant
learners of mathematics
in Australia
who chose not to accept the view of themselves
as
constructed
by their peers or their teacher-and
of others who chose to accept these
constructions.
One student "shouted back" when peers in the mathematics
class shouted
derogatory names.
Discourse is a construct that is wider than mere use of language in conversation (Philips,
1993; Wood, 1998): It embraces all the aspects of social interaction
that come into play when
human beings interact with one another (Dorfler, 2000; Sfard, 2000). The notion of Discourses
formulated by Gee (1992, his capitalization),
and in particular
his extension of the construct to
borderland
Discourses,
those "community-based
secondary Discourses"
situated
in the
"borderland"
between home and school knowledge (p. 146), are in line with Bourdieu's
ideas of
habitus and symbolic violence. Borderland Discourses take place in the borderlands
between
primary (e.g., home) and secondary (e.g., school) cultures of the diverse participants
in social
interactions.
In situations
where the secondary culture (e.g., that of the school) is conceived as
threatening
because of the possibility
of symbolic violence there, the borderland
may be a place
of solidarity
with others who may share a certain habitus. These ideas go some way towards
closing the theoretical
gap in Bourdieu's
characterization
of symbolic violence by raising some
issues of individuals'
choices,
because individuals
choose the extent to which they will
139
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of Mathematics
Education
participate
in various forms of these Discourses
(see also Bishop's,
2002b, use of Gee's
constructs).
Gee's work was in the context of second language learning but is also useful in the
analysis of meanings given to various experiences by mathematics learners in cultural transition
situations.
Bishop (2002a)
used this theoretical
field in moving from the notion of cultural
conflict to that of cultural mediation,
in analyzing these experiences of learners of mathematics.
If one considers the primary Discourse of school mathematics learners to be the home-based
practices
and conversations
that contributed
to their socialization
and enculturation
(forming
their habitus)
in their early years, and continuing to a greater or lesser extent in their present
home experiences, then the secondary Discourse, for the purpose of learning mathematics, could
be designated
as the formal mathematical
Discourse of the established
discipline
of mathematics.
The teacher is more familiar
with this Discourse
than are the students and thus has the
responsibility
of introducing
students to this secondary Discourse. As students become familiar
with this field, their language and practices may approximate more closely those of the teacher.
But in this transition
the borderland
Discourse of interactive
classroom practices provides an
important mediating space.
The enculturating
role of the mathematics
teacher was suggested in the foregoing
account. However, as Bishop (2002b)
pointed
out, the learning
of school mathematics
is
frequently
more of an acculturation
experience than an enculturation.
The difference
between
these anthropological
terms is as follows. Enculturation
is the induction,
by the cultural group,
of young people into their own culture. In contrast, acculturation
is "the modification
of one's
culture through continuous contact with another" (Wolcutt,
1974, p. 136, as quoted in Bishop,
2002b, p. 193). The degree to which the culture of mathematics, as portrayed in the mathematics
classroom, is viewed as their own or as a foreign culture by learners would determine whether
their experiences there would be of enculturation
or acculturation.
(4)
Cultural
Models
Allied
with Gee's (1992)
notion of different
Discourses
is his construct of cultural
models. This construct, defined as '"first thoughts'
or taken for granted assumptions about what
is 'typical'
or 'normal'" (Gee, 1999, p. 60, quoted in Setati, 2003, p. 153), was used by Setati
(2003)
as an illuminating
theoretical
lens in her research on language use in South African
multilingual
mathematics
classrooms.
Already
in 1987,
D'Andrade
defined a cultural
model-which he also called a folk model-as "a cognitive schema that is intersubjectively
shared by a social group," and he elaborated,
"One result of intersubjective
sharing is that
interpretations
made about the world on the basis of the folk model are treated as if they were
obvious facts about the world" (p. 112). The transparency
of cultural
models may help to
explain why mathematics
was for so long considered
to be value- and culture-free.
The
well-known creativity
principle
of making the familiar strange and the strange familiar (e.g., De
Bono, 1975) is necessary for participants
to become aware of their implicit
cultural beliefs and
values, which is why the anthropologist
is in a position
to identify the beliefs that are invisible
to many who are within the culture.
In the context of mathematics learning in multilingual
classrooms, Adler (1998, 2001)
pointed out aspects of the use of language as a cultural resource that relate to the transparency of
cultural models. Particularly
in classrooms where the language of instruction
is an additional
language-not their first language-for
many of the learners (Adler, 2001),
teachers must at
times focus on the language itself, in which case the artifact of language no longer serves as a
"window" of transparent
glass through which to view the mathematical
ideas (Lave & Wenger,
1991). In this case the language used is no longer invisible,
and the focus on the language itself
may detract from the conceptual learning of the mathematical
content (Adler, 200 1 ).
140
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(5)
Cooperation
Valorization
Project Towards the Endogenous Development ofMathematics Education
^
Teachers
'Final Report
in Mathematical
Practices
If transparency
of culture necessitates
making the familiar strange before those sharing
that culture become aware of the lenses through which they are viewing their world, then this
principle
points to a reason for the neglect of issues ofvalorization
in the mathematics education
research community until Abreu's (1993, 1995) research brought the topic to the fore. The value
offormal mathematics as an academic subject was for so long taken for granted that it became a
given notion that was not culturally
questioned.
Especially
in its role as a gatekeeper to higher
education, this status in education
is likely to continue. But if ethnomathematics
as a research
program is to have a legitimate
place in broadening notions both of what counts as mathematics
and of which people have originated
these forms of knowledge, then issues of valorization
assume paramount importance.
Working from the theoretical
fields of cultural psychology
and sociocultural
theory,
Abreu and colleagues
investigated
the effects of valorization
of various mathematical
practices
on Portuguese children
(Abreu, Bishop, & Pompeu, 1997), Brazilian
children
(Abreu, 1993,
1 995), and British children from Anglo and Asian backgrounds (Abreu, Cline, & Shamsi, 2002).
As confirmed also in the research of Gorgorio, Planas, and Vilella
(2002),
many of these
children
denied the existence
of, or devalued, mathematics
as used in practices
that they
associated with their home- or out-of-school settings.
Valorization, the social process of assigning
more value to certain practices than to
others, is closely allied to Wertsch's (1998)
notion of privileging,
defined as "the fact that one
mediational
means, such as social language, is viewed as being more appropriate
or efficacious
than another in a particular
social setting" (Wertsch, 1998, p. 64, in Abreu, 2002, p. 183). Abreu
(2002) elaborated
as follows.
From this perspective,
cultural practices
become associated
with particular
social groups,
which occupy certain positions
in the structure
of society. Groups can be seen as
mainstream or as marginalised.
In a similar vein individuals
who participate
in the practices
will be given, or come to construct, identities
associated
with certain positions
in these
groups. The social representation
enables the individual
and social group to have access to a
'social code' that establishes
relations between practices and social identities,
(p. 184)
Thus Abreu argued strongly that in the learning of mathematics, valorization
operates not only
on the societal plane but also on the personal plane, because it impacts the construction
of social
identities.
At this psychological
level, the construction
of mathematical
knowledge may be
subordinated
to the construction
of social identity
by the individual
learner in cases of cultural
conflict,
as suggested by Presmeg (2002a).
individual
education
(6)
Abreu's ideas are embedded in the field of cultural
psychology.
and society into account, another field that has grown in influence
research in recent decades is that of situated cognition.
Situated
Also taking the
in mathematics
Cognition
The theoretical
field of situated cognition
explores related aspects of the interplay
of
knowledge on the societal and psychological
planes. Hence it has been a useful lens in research
that takes culture into account in mathematical
thinking
and learning
(Watson,
1998). As
Ubiratan D'Ambrosio by his writings
founded and influenced the field of ethnomathematics,
so
Jean Lave in analyzing and reporting her anthropological
research has influenced the theoretical
field of situated
cognition.
From her early theorizing
following
ethnography
with tailor's
apprentices
in Liberia (Lave, 1988) to her more recent writings, following
research with grocery
shoppers and weight-watchers
(Lave, 1997), the notion of transfer of knowledge through
abstraction
in one context, and subsequent
use in a new context, was questioned
and
problematized.
In collaboration
with Etienne Wenger, her theorizing
led to the notion of
141
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of Mathematics
Education
Teachers 'Final
Report
cognitive
apprenticeship
and legitimate peripheralparticipation
(Lave & Wenger, 199 1 ; Wenger,
1998).
In this view, learning consists
in a centripetal
movement of the apprentice
from the
periphery
to the center of a practice, under the guidance of those who are already masters of the
practice.
This theory was not originally
developed
in the context of or for the purpose of
informing
mathematics
education.
However, in its challenge
to the cognitive
position
that
abstract learning of mathematics facilitates
transfer and that this knowledge may be readily
applied
in other situations
than the one in which it was learned (not born out by empirical
research), the theory has been powerful and influential.
The research studies inspired by Lave
and reported by Watson (1998)
bear witness to this strong influence.
In her later writings,
Lave attempted to bring the theory of socially
situated knowledge
to bear on the classroom teaching and learning of mathematics. But issues of intentionality
and
recontextualization
separate
apprenticeship
and classroom
situations,
although
enough
commonality exists in the two situations
for both legitimately
to be called practices (Lerman,
1998).
Mathematics
learners
in school
are not necessarily
aiming
to become either
mathematicians
or mathematics teachers. As Lerman pointed out in connection with the issue of
voluntary
and nonvoluntary
participation,
students'
presence in the classroom
may be
nonvoluntary, creating a very different situation from that of apprenticeship
learning, and calling
into question the assumption
of a goal of movement from the periphery
to the center. At the
same time, the teacher has the intention
to teach her students mathematics,
notwithstanding
Lave's claim that teaching is not a precondition
of learning. The learning of mathematics is the
goal of the enterprise,
and teaching
is the teacher's job. In contrast,
in the apprenticeship
situation
the learning is not a goal in itself, for example, in the case of the tailors the goal is to
make garments efficiently.
Thus, as Adler (1998)
suggests,
"It is in the understanding
of the
aims of school education that Lave and Wenger's seamless web of practices entailed in moving
from peripheral
to full participation
in a community of practice is problematic"
(p. 174).
Although the theory of legitimate
peripheral
participation
may not translate easily into
the classroom teaching and learning of mathematics,
the view of learning as a social practice has
powerful implications
for this learning and has been an influence in changes that have taken
place in the practice of teaching mathematics,
such as an increased emphasis on communication
(NCTM, 1989, 2000) in the mathematics classroom. Even more, situated cognition
as a theory
has given a warrant to attempts to bridge the gap between in- and out-of-school
mathematical
practices.
(7)
Use of Semiotics
in Linking
Out-Of-School
and In-School
Mathematics
In the USA, the Principles
and Standards for School Mathematics (NCTM, 2000) continued
the earlier call (NCTM, 1989) for teachers to make connections,
in particular
between the
everyday practices
of their students
and the mathematical
concepts that are taught in the
classroom. But various theoretical
lenses of situated cognition
(Lave & Wenger, 1991 ; Kirshner
& Whitson, 1997) remind us that these connections can be problematic.
There are at least three
ways in which the activities
of out-of-school
practices differ from the mathematical
activities
of
school classrooms (Walkerdine,
1988), as follows.
>
The goals of activities
>
Discourse
patterns
in the two settings
of the classroom
differ
radically.
do not mirror those of everyday
>
practices.
Mathematical
terminology
and symbolism have a specificity
that differs markedly from
the useful qualities
of ambiguity and indexicality
(interpretation
according to context)
of terms in everyday conversation.
A semiotic framework that uses chains of signification
(Kirshner
& Whitson, 1997) has the
potential
to bridge this apparent gap through a process of chaining of signifiers
in which each
sign "slides under" the subsequent signifier. In this process, goals, discourse patterns, and use of
142
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Project
of Mathematics
Education
terms and symbols all move towards that of classroom mathematical
practices in a way that has
the potential
to preserve essential structure and some of the meanings of the original activity.
This theoretical
framework resonates with that of Realistic
Mathematics Education
(RME) developed
by Freudenthal,
Streefland,
and colleagues
at the Freudenthal
Institute
(Treffers,
1993).
Realistic
in this sense does not necessarily
mean out-of-school
in the real
world:
The term refers to problem
situations
that learners
can imagine
(van den
Heuvel-Panhuizen,
2003). A theory of semiotic chaining
in mathematics
education resonates
with RME in that the starting points for the learning
are realistic
in this sense. But more
specifically
the chaining model is a useful tool for linking out-of-school
mathematical
practices
with the formal mathematics of school classrooms (an example of such use is presented
in the
next paragraph).
In brief, the theory of semiotic chaining used in mathematics education research, as it
was initially
presented by Whitson (1997),
Cobb, Gravemeijer,
Yackel, McClain, & Whitenack
(1997),
and Presmeg (1998b),
followed
the usage of Walkerdine
(1988).
Although
she was
working in a poststructural
paradigm,
Walkerdine
found the Swiss structural
approach of
Saussure, as modified by Lacan, useful in building
chains of signifiers
to link a home practice,
such as that of a daughter and her mother pouring drinks for guests, with more formal learning
of mathematics, in this case the system of whole numbers. Walkerdine's
chain had the following
structure.
Signified
1: the actual
S kirrrtlfiot*
1 (vtrrnsJiYirr
J. \oiL*rii*i./tgjur
^
å AIXA\Î
The daughter
is asked
people
coming to visit;
å £-»*•E cirmi-fitzA
oigiiinû
~\ \'
4-\^<c%
r\r%f
lj.
uio
named
i
-X-f +T-lrt «AAM1rt
}å
Signl
f
Sign2
^
Sign
ui UlC pcupic.
to raise a finger for each name.
Signified
2: the names of the euests:
dignuier
l: tne raised
lingers.
The daughter is asked to count her raised
Signified
3: the raised fingers;
fingers.
Signifier
four, five.
3: the numerals,
one two, three,
3
In this process, signifier 3 actually comes to stand for all of the preceding links in the chain:
Five guests are coming to tea. Note that each signifier in turn becomes the next signified,
and
that at any time any of the links in the chain may be revisited
conceptually.
As mother and
daughter move through the three signs, the discourse shifts successively
from actual people to
their names, to the fingers of one hand, and finally to the more abstract discourse of the
numerals of the whole number counting system.
In her distinctive
process as that of crossing
style, Adler (1998)
a discursive
bridge:
characterized
the associated
recontextualizing
That there is a bridge to cross between everyday and educated discourses is at the heart of
Walkerdine's
(1988)
argument for 'good teaching'
entailing
chains of signification
in the
classroom where everyday notions have to be prised out of their discursive
practice and
situated in a new and different discursive practice, (p. 174)
Using this process, a teacher can use chains of signification
as an instructional
model that
develops a mathematical
concept starting with an out-of-school
situation
and linking
it in a
number of steps with formal school mathematics (Hall, 2000). Building
on the work of Presmeg
143
International
Cooperation
Project
Education
( 1 998b) in his dissertation
research, Hall taught three 4th-grade teachers to build semiotic chains
appropriate
to the practices
of their students and the instructional
needs of their individual
classrooms. Using a similar semiotic
chaining
model, Cobb et al. (1997)
reported
on the
emergence (using their emergent theoretical
perspective)
of chains of signification
in one
lst-grade
classroom.
In addition
to these sources, examples of mathematical
chaining
at
elementary school, high school, and college levels may be found in Presmeg's (1998b,
2006)
writings.
In later research on building
bridges between out-of-school and in-school mathematics,
Presmeg (2006)
preferred to use a nested triadic model based on the writings of Charles Sanders
Peirce (1992,
1998).
In addition
to an object (which could be an abstract concept)
and a
representamen that stands for the object in some way, each nested sign has a third component
that Peirce called the interpretant.
This triadic model explicitly
allows for learners' individual
construction
of meaning (through
the interpretant)
in a way that linear chains of signification
can do only implicitly.
Attempts to facilitate
teachers'
their classrooms
provide examples
continued in the next section.
2.3
Empirical
Fields
That
use of children's
out-of-school
of practical
uses of theoretical
Incorporate
Culture
mathematical
practices in
fields. This theme is
in Mathematics
Education
The last section dealt with some theoretical
fields that have been useful, or have the
potential
to be useful, in research on the role of culture in mathematics education. In the current
section,
some associated
empirical
fields are discussed.
Empirical
fields entail
broad
methodologies
of empirical
research, but in addition
to these, in this section specific methods of
research and also the participants,
data, and results of selected studies are introduced.
These
details of participants,
time, and place in the research are termed the empirical settings (Brown
& Dowling, 1998). Thus appropriate
empirical
fields and settings are the focus of this section,
along with some results of research on the role of culture in mathematics education.
Because ethnography
is the special
province of holistic
cultural
anthropological
research (Eisenhart,
1988), which has as a broad goal the understanding
of various cultures, it is
natural to expect that ethnography
(sometimes in modified form) would be used by researchers
who are interested in the role of culture in teaching and learning mathematics. This is indeed the
case. However, researchers in mathematics
education in general sometimes call their studies
ethnography
when they have not spent long enough in the field to warrant that term for their
research (Eisenhart,
1988). For instance, valuable as studies such as that ofMillroy
(1992)
are
for mathematics
educators to learn more about the use of mathematical
ideas in out-of-school
settings
such as that
of Millroy's
carpenters
in Cape Town, South
Africa,
the 5y^
month
duration of her fieldwork might be considered
brief for an ethnography,
and most ethnographic
studies in mathematics
education have a shorter duration than Millroy's.
For this reason, the
field of such research is often more appropriately
called case studies (Merriam, 1998), because
they satisfy the criterion
of being bounded in particular
ways. If such studies are investigating
a
particular
topic, such as ways of bringing
out-of-school
mathematics into the classroom, and
using several cases to do so, they could be called instrumental
case studies (Stake, 2000).
Where the individual
cases themselves are the focus, case studies are intrinsic (ibid.).
Most of the studies mentioned in this section may be regarded as instrumental
case
studies
in the sense explained
in the foregoing.
Research that concerns in-school
and
out-of-school
mathematical
practices
falls into two main categories.
In the first category are
studies that attempt to build bridges
between informal or nonformal mathematical
practices
(Bishop,
Mellin-Olsen,
& van Dormolen, 1991) and those of formal school mathematics. In the
second category are studies that link with formal mathematics education less directly
but with
the potential
to deepen understanding
of this education, by exploring the culture of mathematics
144
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Teachers 'Final
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in various workplaces. The studies in both of these categories
are too numerous for a summary
chapter such as this one to do full justice
to the aims, research questions,
methodologies
and
methods for data collection,
and the subtleties
unearthed in the results of such research. (Again,
the reader is reminded to read the original
literature
for depth of coverage.) What follows is an
introduction
to the range of research in each of these two categories, with discussion
of some of
the empirical
settings, a sampling of issues addressed, and some of the results.
2.4
Linking
Mathematics
Learning
Out-Of-School
and In-School
In the first category, studies that specifically
investigated
linking
out-of-school
and
in-school mathematics
are of several types. Some researchers interviewed
parents of minority
learners and their teachers to investigate
relationships
between home and school learning of
mathematics. In England, Abreu and colleagues
conducted such interviews for the purpose of
investigating
how the parents of immigrant learners are able to support their children
in their
transitions
to learning mathematics in a new school culture (Abreu, Cline & Shamsi, 2002).
Civil and colleagues
in the USA interviewed
such parents for the purpose of finding out which
home practices
of immigrant
families
might be suitable
for pedagogical
purposes
in
mathematics classrooms (Civil,
1990, 1995, 2002; Civil & Andrade, 2002). Related empirical
studies in which researchers investigated
the transitions
experienced
by immigrant children
in
their learning
of mathematics
have been reported by Bishop (2002a)
in Australia
and by
Gorgorio, Planas, and Vilella (2002)
in Catalonia,
Spain.
In the next type of study in this category, research was conducted in mathematics
classrooms to investigate
the effects of bringing
out-of-school
practices into that arena. Using
video recording as a data collection
method, Brenner (2002)
investigated
how four teachers and
their junior high school students went about an activity in which the students were required to
decide cooperatively
which of two fictitious
pizza companies should be given the contract to
supply pizza to the school cafeteria.
In a different
classroom video study, Moschkovich's
(2002b)
research addressed
the mathematical
activities
of seventh-grade
students during an
architectural
design project: Students working collaboratively
tried to design working and living
quarters for a team of scientists
in Antarctica
in such a way that space would be maximized
while still taking into account the heating costs of various designs. These studies emphasized the
point that through their pedagogy and instructional
decisions
in the classroom, teachers are an
important
component of the success (or lack of success) of attempts to use out-of-school
practices
effectively
in mathematics
classrooms.
Beliefs
of teachers
about the nature of
mathematics, what culture is, and its role in the classroom learning of mathematics are strong
factors in teachers' decisions
in this regard (Civil,
2002; Presmeg, 2002b).
Like the researchers in the preceding
paragraphs,
Presmeg (1998b,
2002b, 2006) and
Hall (2000)
recognized
the lack of congruence of out-of-school
cultural
practices and those
"same" practices
when brought into the school mathematics
classroom (Walkerdine,
1988).
Their research used semiotic chaining as a theoretical
tool (see previous section) in an attempt
to bridge the gaps, not only between these practices
in- and out-of-school,
but also between
mathematical
ideas implicit
in these activities
and the formal mathematics of the syllabi
that
teachers are expected to use in their practices.
The fourth-grade
teachers with whom Hall
worked built chains of signifiers
that they implemented
with their classes, starting
with a
cultural practice of at least some of their students and aimed at linking
mathematical
ideas in
this practice in a number of steps with a formal school mathematics topic. Chains constructed by
the teachers
were designated
as either intercultural-bridgmg
two or more cultures,
or
intracultural-hav'mg
a chain that remained within a single culture. Examples of the first type
involved number of children
in students' families,
pizzas, coins, and measurement of students'
hands, linking in a series of steps with classroom mathematical concepts. These are intercultural
because the cultures and discourse of students' homes or activities
are linked with the different
discourse
and culture of classroom mathematics,
for instance,
the making of bar graphs.
145
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of Mathematics
Education
Manipulatives
were frequently
used as intermediate
links in these chains. The intracultural
type,
involving
chains that were developed within the culture of a single activity,
was evidenced in a
chain involving
baseball
team statistics.
The movement along the chain could be summarized as
follows:
BaseballGame
-åº Hitsvs.AtBats
-åº SuccessFraction
-åº BattingAverage.
It was not the activity
that was preserved throughout
the chain, but merely the culture of
baseball
(at least as it was imported into the classroom practice)
within which the chain was
developed.
The need for interpretation
at each link in the chain led to further theoretical
developments
using a triadic nested model that was a better lens for interpreting
the results
(Presmeg, 2006).
One study that is ethnographic
in its scope and methodology has less explicit
claims to
link out-of-school
and in-school mathematics, although implicit
ties between the two contexts
are present. This study is a thorough investigation
of mathematical
elements in learning the
practice of selling newspapers in the streets by young boys (called in Portuguese ardinas) on the
island of Cabo Verde in the Atlantic ocean (Santos & Matos, 2002). Using Lave and Wenger's
( 1 99 1 ) theoretical
framework of legitimate
peripheral
participation,
Santos and Matos described
the goal of their research as follows.
Our goal was to look into the ways (mathematics)
learning relates to forms of participation
in social practice
in an environment where mathematics is present but that escapes the
characteristics
of the school environment. Because we believe that culture is an unavoidable
fact that shapes our way of seeing and analyzing things, we decided to look at a culturally
distinct
practice
and that constituted
a really strange domain for us: the practice
of the
ardinas at Cabo Verde islands in Africa, (p. 81)
In addition to interesting
explicit
and implicit
mathematical
aspects of the changing practice
of the ardinas as they moved from being newcomers to full participants,
methodological
difficulties
in this kind of research were foregrounded by Santos and Matos (2002):
The fact that the research is studying a phenomena [sic] which was almost totally strange to
us in most of its aspects, led us to realize that we had to go through a process which should
involve, to a certain extent, our participation
in the (ardinas ') practice with the explicit
(for
us and for them) goal of learning it but not in order to be a full member of that community
of practice. This starting point (more in terms of knowing that there are more things that we
don't know than that we know) opened that community of practice to us but also gave us
consciousness that methodological
issues were central in this research, (p. 120)
Many of the difficulties
that Santos and Matos described
with sensitivity
and vividness
are
commonto most anthropological
research and thus also impact investigations
of cultural
practices
in the field of mathematics
education. They were required to become part of the
culture sufficiently
to be able to interpret
it to others who are not participants,
but at the same
time not to become so immersed in the culture that it would be transparent-a
lens that is not the
focus of attention because one looks through it. They faced the difficulty
that entering the group
as an outsider might in fact change the culture of that group to a greater or lesser extent. On
several occasions there was evidence that the mathematical
reporting by key informants among
the ardinas was influenced by the fact that the fieldworker was a Portuguese-speaking
woman,
whom these boys might have associated
with their schoolteachers.
Finally,
the researchers
recognized the not inconsiderable
difficulties
associated
with practical
matters concerned with
collecting
data in the street.
The investigation
of the ardinas' mathematical
thinking
could be classified
as an
ethnographic
workplace study. Thus this short account of this research provides a transition
to
the second aspect of this empirical
section of the chapter, mathematics in the workplace.
(1)
Culture
of Mathematics
In the second category
in the Workplace
of studies
in this
146
section,
the culture
of mathematics
in various
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Education
workplaces was investigated
intensively
by FitzSimons
(2002),
by Noss and colleagues (Noss &
Hoyles, 1996a; Noss, Hoyles & Pozzi, 2000; Noss, Pozzi & Hoyles, 1999; Pozzi, Noss, and
Hoyles, 1998), and by several other researchers (see later). FitzSimons
investigated
the culture
and epistemology
of mathematics in Technical and Further Education in Australia,
and how this
education related or failed to relate to the cultures of mathematics as used in workplaces. Noss,
Hoyles, and Pozzi examined workplace mathematics
in more specific detail:
Inter alia their
studies addressed mathematical
aspects of banking and nursing practices.
A commonthread in
these studies is the demathematization
of the workplace: "As the seminal work of researchers
Richard Noss and Celia Hoyles among others indicates,
mathematics
actually
used in the
workplace is contingent
and rarely utilizes
'school mathematics'
algorithms
in their entirety,
if
at all, or necessarily
correctly" (FitzSimons,
2002, p. 147).
The same point was brought out strongly,
but contingently,
in an investigation
of
mathematical
activity
in automobile
production work in the USA (Smith, 2002). One result of
Smith's study was that "the organizational
structure and management of automobile
production
workplaces directly
influenced
the level of mathematics
expected of production
workers" (p.
112).
This level varied from a minimal expectation
on assembly
lines-designed
to be
"worker-proof'-to
"a surprisingly
high level of spatial
and geometric competence, which
outstripped
the preparation
that most K-12 curricula provide" (p.1 12), e.g., as manifested
by
skilled
machinists
who translated between two- and three-dimensional
space with sophistication
and accuracy in creating products that were sometimes one-of-a-kind. Organization
that used
"lean manufacturing
principles"
(p. 124) to move away from assembly lines and give more
autonomy to workers was more likely to enhance and require these highly developed
forms of
mathematical
thinking.
Workers are not usually allowed to decide what level of mathematics
will be required in their work:
The fact that the organization
of production
systems and work practices mediates and in
some cases limits workers' mathematical
expectations
and their access to workplace
mathematics is a reminder that the everyday mathematics of work is inseparable
from issues
of power and authority.
In large measure, someone other than the production
workers
themselves
decides
when, how often, and how deeply they will be called on to think
mathematically.
(Smith, 2002, p. 130)
Other sources of insights
into the mathematical
ideas involved in practices
of various
workplaces are detailed
in FitzSimons's
(2002) book. These include the following:
operators in the light metals industry (Buckingham,
1997); front-desk
motel and airline staff
(Kanes, 1997a, 1997b);
landless
peasants in Brazil...(Knijnik,
1996, 1997, 1998);
carpet
layers (Masingila,
1993);
commercial
pilots
(Noss,
Hoyles,
& Pozzi, 2000);
...
draughtspersons
(Strasser,
1998);
semi-skilled
operators (Wedege, 1998b, 2000a, 2000b)
and swimming pool construction
workers (Zevenbergen,
1996). Collectively,
these reports
highlight
not only the breadth and depth of mathematical
concepts encountered in the
workplace,
but underline
the complex levels of interactions
in the broad range of
professional
competencies
as outlined above, where mathematical
knowledge can come into
play - when permitted by (or in spite of) management, (p. 72)
In addition
to this broad range of reported research, in an early study Mary Harris
(1987)
investigated
"women's work," challenging
her readers to "derive a general
expression for [knitting]
the heel of a sock" (p. 28). The more recent mathematics education
conferences have included presentations
on the topic of the mathematics of the workplace
(e.g., Strasser, 1998).
From the foregoing,
the breadth of this field, the variety of practices
that have been
investigated
from a mathematical
point of view, and the scope of methodologies
employed are
apparent. Many of these studies may be characterized
as investigations
in ethnomathematics,
for
example, those of Masingila
(1993)
with the art of carpet laying,
and Harris (1987).
This
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Education
overlap in classification
also applies to research on mathematics in the world of work that is
concerned with its artifacts
and tools, and with its technology.
The confluence of technology and
culture in mathematics education is the topic of the last part of this section.
(2)
Influences
of Technology
A study that stands at the intersection
of workplace mathematics studies and those that
are concerned with the role and influence
of technology
in mathematics
education is that by
Magajna and Monaghan (2003).
In their research they investigated
both the mathematical
elements and the use of computer aided design (CAD) and manufacture (CAM) in the work of
six skilled
technicians
who designed and produced moulds for glass factories. The technicians
specialized
in "moulds for containers of intricate
shapes" (p. 102), such as bottles in the form of
twisted pyramids, stars, or guitars. The study is interesting
because the mathematics used by the
CAD/CAM technicians
in various stages of their work (for instance, calculating
the interior
volume of a mould) was not elementary, and in theory school-learned
mathematics could have
been used. However, the researchers reported as follows.
Although the technicians
did not consider their activity
was related to school mathematics
there is evidence that in making sense of their practice they resorted to (a form of) school
mathematics. The role of technology
in technicians'
mathematical
activity
was crucial: not
only were the mathematical
procedures they used shaped by the technology
they used but
the mathematics
was a means to achieve technological
results.
Further to this the
mathematics
employed by the technicians
must be interpreted
within the goal-oriented
behaviour of workers who ' live' the imperatives
and constraints
of the factory's production
cycle, (p. 101)
An implication
of results such as these, resonating
with the conclusions
of other workplace
studies,
is that there is no direct path from school mathematics to the mathematics used,
sometimes indirectly,
in various occupations,
where the specific practices are learned on the job.
It may not be possible
then to gear a mathematics curriculum, even in vocational
education
(FitzSimons,
2002),
to the specific
requirements
of a number of vocations simultaneously.
However, as technology
has developed
in the last 6 decades, its influence
has been felt in
mathematics curricula of various time periods, as illustrated
by Kelly (2003).
The research in this growing field is beyond the scope of this paper, but an example
highlights
the potential
of graphing calculators
and computers to change the culture of learning
mathematics.
The research of Ricardo Nemirovsky (2002)
and his colleagues
(Nemirovsky,
Tierney, & Wright,
1998)
demonstrates
how the use of motion detectors
and associated
technology
changes the culture of learning from one of fostering formal generalizations
in
mathematics
(e.g., "all x are /'), to one of constructing
situated generalizations.
The latter kind
of generalization
is "embedded in how people relate to and participate
in tasks, events, and
conversations"
(Nemirovsky,
2002, p. 250)-and
it is "loaded" with the values of"grasping
the
circumstances and transforming
aspects that appear to be just ordinary or incidental
into objects
of reflection
and significance"
(p. 25 1). Nemirovsky
illustrated
this kind of generalization
in the
informal mathematical
constructions
of Clio, an 1 1-year-old
girl working with the problem of
trying to predict, based on the graph created on a computer screen by a motion detector, where a
toy train is situated
in a tunnel.
In summary, then, technology,
by entering all avenues of life, influences
not only the
mathematics of the curriculum and the ontology of mathematics itself, but also the culture of the
future, through its children.
This section has examined in broad detail some of the empirical
research relating to the
role of culture in teaching and learning mathematics. Aspects that were considered included the
linking
of out-of-school
and in-school mathematics
learning,
both from the point of view of
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comparing these different
cultural
practices
and from the point of view of building
bridges
between cultural practices and the classroom learning of mathematics-of
"bringing in the world"
(Zaslavsky,
1996). A related strand that was outlined was the culture of mathematics
in the
workplace, and finally, some aspects of the cultural influence of technology
on the learning of
mathematics were suggested. Clearly in all of these strands ongoing research is needed and in
progress.
However, perhaps just as significantly,
research
is needed that will increase
understanding
of and highlight
how these strands are related amongst themselves. For instance,
as the use of technology
both in the workplace and in the mathematics classroom changes the
culture of learning mathematics in these broad arenas, hints of avenues to be explored in future
work appear, for example in whether and how technology
has the potential
to bridge or decrease
the gap between formal and informal mathematical
knowledge. The theme of technology
and
culture in future research is elaborated,
along with other significant
areas, in the final section.
2.5
Future
Directions
for Research
on Culture
in Mathematics
Education
The final section uses a wider lens to zoom out and consider areas of research on culture
in mathematics
education
that require
development
or are likely
to assume increasing
importance in the years ahead. By its nature, this section is speculative,
but trends are already
apparent that are likely to continue. The influence of technology
is one such trend (Morgan,
1994; Noss, 1994).
(1 )
Technology
Already
in 1988, Noss pointed
out the cultural
entailments
of the computer in
mathematics
education,
as follows: "Making sense of the advent of the computer into the
mathematics
classroom entails a cultural perspective,
not least because of the ways in which
children
are developing
the computer culture by appropriating
the technology
for their own
ends" (p. 251). He elaborated,
"The key point is that children see computer screens as 'theirs',
as a part of a predominantly
adult culture which they can appropriate
and use for their own
ends" (p. 257). As these children become adults with a facility with technology
beyond that of
their parents (Margaret Mead's prefigurative enculturation
comes into play as children
teach
their parents),
the culture of mathematics education in all its aspects is likely to change in
fundamental ways. Noss (1988)
and Noss and Hoyles (1996b)
put forward a vision for the role
that computer microworlds have played and might play in the future of mathematics education.
With accelerating
changes in platforms, designs, and software, research must keep pace and
inform resulting
changes in the cultures of teaching and learning of mathematics in schools. An
aspect of potential
use for the computer stems from its ability
to mimic reality in a world of
virtual
reality.
Noting the complex relationship
between formal and informal mathematics
(ethnomathematics),
Noss ( 1 988) suggested,
"I propose that the technology
itself-specifically
the computer-can be the instrument for bridging the gap between the two" (p. 252). This area
remains one in which research is needed, both in its own right and in ways that the change of
context of bringing the virtual reality of computer images into mathematics classrooms changes
the culture of teaching and learning in those classrooms.
(2)
Bridging
Mathematics
In-School
and Out-Of-School
Resonating with the Realistic
Mathematics Education viewpoint that real contexts are
not confined
to those concrete situations
with which learners
are familiar
(van den
Heuvel-Panhuizen,
2003), Carraher and Schliemann (2002) made a useful distinction
between
realism and meaningfulness:
What makes everyday mathematics powerful
everyday realism of the situations,
but the
consideration
(Schliemann,
1995). In addition
realism (D. W. Carraher & Schliemann, 1991).
149
is not the concreteness of the objects or the
meaning attached
to the problems
under
meaningfulness
must be distinguished
from
It is true that engaging in everyday activities
International
Cooperation
Project
Development
of Mathematics
Education
such as buying and selling,
sharing, or betting may help students establish
links between
their experience and intuitions
already acquired and topics to be learned in school. However,
webelieve it would be a fundamental mistake that schools attempt to emulate out-of-school
institutions.
After all, the goals and purposes of schools are not the same as those of other
institutions,
(p. 137)
The dilemma, then, is how to incorporate
out-of-school
practices
in school mathematics
classrooms in ways that are meaningful to students and that do not trivialize
the mathematical
ideas inherent in those practices.
This issue remains a significant
one for mathematics education
research.
Reported research (Civil,
2002; Masingila,
2002; Presmeg, 2002b)
has shown that
learners' beliefs
about the nature of mathematics
affect what mathematics
they identify
as
mathematics in out-of-school
settings.
Students' and teachers' conceptions
of mathematics
as
decontextualized
and abstract may limit what can be accomplished
in terms of meaningfulness
derived from out-of-school
practices
(Presmeg,
2002b).
And yet, abstract thinking
is not
necessarily
antagonistic
to the idea of reasoning
in particular
contexts, as Carraher and
Schliemann
(2002)
pointed
out, which led them to propose the construct
of situated
generalization.
They summarized these issues as follows.
Research sorely needs to find theoretical
room for contexts that are not reducible
to physical
settings
or social structures to which the student is passively
subjected.
Contexts can be
imagined, alluded to, insinuated,
explicitly
created on the fly, or carefully
constructed over
long periods of time by teachers and students. Much of the work in developing
flexible
mathematical
knowledge depends on our ability to recontextualize
problems - to see them
from diverse and fresh points of view and to draw upon our former experience, including
formal mathematical
learning.
Mathematization
is not to be opposed to contextualization,
since it always involves thinking
in contexts. Even the apparently
context-free activity
of
applying
syntax transformation
rules to algebraic
expressions
can involve meaningful
contexts, particularly
for experienced mathematicians,
(p. 1 47)
They mentioned the irony that the
approaches of both highly unsuccessful
mechanical
and highly
following
successful
of algorithms
characterizes
mathematical
thinkers.
the
Taking into account both the need for meaningful
contexts in the learning
of
mathematics and the necessity
of developing
mathematical
ideas in the direction
of abstraction
and generalization
(in the flexible sense, not to be confused with decontextualization),
at least
two extant fields of research have the potential
to address these issues in significant
ways. The
notions of horizontal
and vertical mathematizing
that have informed Realistic
Mathematics
Education
research for several decades (Treffers,
1993)
could resolve a seeming conflict
between abstraction
and context. In harmony with these ideas, recent attempts to use semiotic
theories
in linking
out-of-school
and in-school mathematics also have the potential
for further
development (Hall, 2000; Presmeg, 2006).
As Moschkovich
(1995)
pointed
out, a tension exists between educators'
attempts to
engage learners
in "real world" mathematics
in classrooms
and movements to make
mathematics
classrooms
reflect the practices
of mathematicians.
Multicultural
mathematics
materials
for use in classrooms have been available
for some time (Krause, 1983; Zaslavsky,
1991,
1996).
However, the tensions,
the contradictions,
and the complexity
of trying to
incorporate
practices
for which "making change" serves as a metonymy at the same time that
students
are "making mathematics"
(Moschkovich,
1995)
will
engage
researchers
in
mathematics education for some time to come. Bibliographies
such as that compiled by Wilson
and Mosquera (1991)
will continue to be necessary, to inform both researchers and practitioners
what has already been accomplished
as the field of culture in mathematics
education continues
to grow.
150
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(3)
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Teacher
Project
of Mathematics
Education
Education
In all of the foregoing areas of potential
cultural research in mathematics education, the
role of the teacher
remains important.
Noting
that concrete and abstract
domains in
mathematical thinking
are not necessarily
disparate,
Noss (1 988) suggested,
The key idea is that of focusing attention on the important relationships
involved, a role in
which-as Weir (1987)
points out-the computer is rather well cast; but not without the
conscious intervention
of educators, and the careful development of an ambient learning
culture, (p. 263)
Bishop
(1988a,
1998b)
was also intensely
aware of the role of the mathematical
enculturators
in personifying
the values that are inherent in the teaching
and learning
of
mathematics. In the final chapter of his seminal book (1988a)
he suggested requirements
for the
education-rather
than the more restricted
notion of "training"-of
those who will be mathematics
teachers, at both the elementary and secondary levels. He did not distinguish
between these
levels,
for teachers
at both elementary
school and secondary school have important
mathematical
enculturation
roles. Bishop ( 1 988a) summarized the necessary criteria as follows.
I propose,
then, these four criteria
-ability
for the selection
to 'personify'
the mathematical
-commitment to the Mathematics
-ability
to communicate
- acceptance
of suitable
ofaccountability
enculturators:
culture
enculturation
Mathematical
Mathematical
process
ideas and values
to the Mathematical
cultural
group, (p. 168)
These ideas still seem timely;
in fact the literature
on discourse
and communication
has
broadened in the decades since these words were written, to suggest that communication
amongst all involved in the negotiation
of the cultures of mathematics classrooms (teacher and
learners)
plays a significant
role in the learning of mathematics in those arenas (Cobb et al.,
1997; Dorfler, 2000; Sfard, 2000). Language and Communication in Mathematics Education
was the title of a Topic Study Group at the 10th International
Education
(Copenhagen,
2004), and the literature
in this field is already extensive. But how to
educate future teachers of mathematics to satisfy Bishop's
four criteria
is still an open field of
research.
As Arcavi (2002)
acknowledged,
much has already been accomplished
in curriculum
development,
research on teachers' beliefs and practices,
and "the development of a classroom
culture that functions in ways inspired by everyday practices of academic mathematics" (p. 27).
However, open questions still exist concerning ways of using the recognition
that the transition
from out-of-school
mathematical
practices
to those within
school
is sometimes
not
straightforward,
in order to inform the practices of mathematics teaching.
Arcavi (2002)
gave
examples of such questions.
However, much remains to be researched. For example, is it always possible
to smooth the
transition
between familiar and everyday contexts, in which students use ad hoc strategies
to
solve problems, and academic contexts in which more general, formal, and decontextualized
mathematics is to be learned? Are there breaking points? If so, what is their nature? Studies
in everyday mathematics and in ethnomathematics
are very important contributions,
not
only because of their inherent value but also because of the reflection
they provoke in the
mathematics
education
community at large. There is much to be gained from those
contributions,
(p. 28)
already
Clearly
ethnomathematics
conceived
permeated the cultures of mathematics
as a research program (D'Ambrosio,
2000) has
education research and practice in various ways.
151
International
Cooperation
The influence,
are ongoing.
Project
and the need for research
Development
that addresses
ofMathematics
Education
Teachers 'Final
the complexities
of the issues
Report
involved,
Acknowledgements
The author wishes to thank Ubiratan D'Ambrosio and Joanna Masingila
for helpful comments
on an early draft of the Handbook chapter on which this paper is based, and Ubiratan
D'Ambrosio for supplying
the list of useful web sites, including
those of relevant dissertations.
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The relationship
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(pp. 491-511).
Dordrecht,
The
Netherlands:
Kluwer Academic.
Sfard,
A. (2000).
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reality
into being - or how mathematical
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mathematical
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and communicating
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classrooms:
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activity
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& J. N. Moschkovich
(Eds.),
Everyday and academic mathematics
in the classroom (pp.
1 1 1-130).
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National
Stake,
R. E. (2000).
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In N. K. Denzin & Y. S. Lincoln
research (pp. 435-454).
Thousand Oaks, CA: Sage.
Steen,
L. A. (1990).
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L. (1 967).
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Education:
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in Mathematics,
31(1-2),
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and mythbreakiing
151-173.
Proto-mathematics
Thompson, A. G. (1984).
The relationship
teaching
to instructional
practice.
A. (1993).
New approaches
Handbook
to numeracy.
of qualitative
Washington,
D. C:
& Talley.
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for work: A didactical
perspective.
In C. Alsina,
J. M. Alvarez, B.
C. Laborde,
& A. Perez (Eds.),
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Selected lectures (pp. 427-441).
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Spain: S. A. E. M 'THALES'.
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16(2),ll-18.
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of giants:
(Eds.),
Wiskobas
in the mathematics
and/or
real mathematics.
classroom.
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Studies
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of teachers'
conceptions
of mathematics
and mathematics
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Studies in Mathematics,
15, 1 05-127.
and Freudenthal
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158
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Mathematics,
Project
The didactical
use of models in Realistic
trajectory
on percentage. Educational
Vithal,
R. (2003).
Towards a pedagogy of conflict
The Netherlands
: Kluwer Academic.
Vithal,
R. & Skovsmose, O. (1 997). The end of innocence:
Studies in Mathematics, 34(2), 13 1-157.
and dialogue
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New York: Routledge.
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(1998).
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cognition
and the learning
of Oxford Department of Educational
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Wenger, E. (1998).
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Communities of practice:
University
Press.
Whitson,
Mind as action.
Oxford:
Oxford
education.
Educational
and the production
of mathematics.
meaning, and identity.
University
Dordrecht,
of 'ethnomathematics.'
developments
Learning,
Mathematics
Education:
An
Studies in Mathematics, 54(1),
for mathematics
Watson, A. (Ed.)
University
Wertsch, J. V. (1 998).
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(2003).
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9-35.
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of rationality.
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Cambridge,
England:
England:
Press.
J. A. (1997).
Cognition
as a semiosic
process:
From situated
mediation
to critical
reflective
transcendance.
In D. Kirshner
& J. A. Whitson (Eds), Situated
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Social, semiotic, and
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(pp. 97- 1 49). Mahwah, NJ: Lawrence Erlbaum Associates.
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mathematics
across culture and time. In H. Selin (Ed.), Mathematics
across cultures:
The history
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(pp. 1-12).
Dordrecht,
The
Netherlands:
Kluwer Academic.
Zaslavsky,
C. (1973).
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Africa
counts:
Zaslavsky,
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32-36.
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Zaslavsky,
C. (1 996).
Heinemann.
The multicultural
Useful
web sites on culture
cultures
Number andpattern
in the mathematics
mathematics
in African
class.
classroom:
and the learning
cultures.
For the Learning
Bringing
and teaching
Lawrence
of Mathematics
in the world. Portsmouth,
of mathematics:
http ://www.csus.edu/indiv7o/oreyd/once/once.htm
http ://www. ge o metry. n et/pure_and_app l ied_m ath/emnom athemat i c s. htm l
http://www.dm.unipi.
it/~jama/ethno/
http://phoenix.
sce.fct.unl.pt/GEPEm/
http ://www. fe. unb. br/etnom atemat ica/
http://www2.fe.usp.br/~etnomat
http://web.nmsu.edu/~pscott/spanish.htm
http://etnomatematica.unival
le.edu.co
http ://www.rpi. edu/~eglash/isgem.htm
http ://chronicle.com/colloquy/2000/ethnomath/ethnomath.htm
http ://chronicle.com/free/v47/i06/06aO
1 60 1.htm
http://chronicle.com/colloquy/2000/ethnomath/re.htm
http ://www. ecsu. ctstateu. e du/depts/edu/proj
ects/ethnom ath. htm l
159
Chicago:
Hill
ll(2)
NH:
International
3.
Cooperation
Project
Teaching
for numeracy
: Re-examining
the role of cultural
aspects
of Mathematics
Education
Teachers 'Final
Report
of mathematics
WeeTiong Seah
Monash University
3.1
Numeracy
in the school
mathematics
curriculum
Developments and advancement - scientific / technological
or otherwise - in the world we live
in have meant that over the years, we acquire or adapt new knowledge, new skills, new ways of
doing the same tasks, new outlooks, new attitudes, and new perspectives.
Likewise, the purpose
of formal school education in general, and also institutional
aims for the teaching and learning
of mathematics in schools in particular,
have evolved in new forms in response to new
sociocultural
demands. For school mathematics, one of the significant
moves in the early 2000s
in countries such as Australia,
New Zealand, Sweden and United Kingdom is towards the
teaching of mathematics for numeracy. This move acknowledges that general education for all
implies that not all school students need to be trained to be mathematicians.
Rather, general
education for the masses has provided nations with the opportunity
to educate their respective
citizens so that they acquire the capacity to cope with the evolving changes in related demands
in daily life. This is generally
observed in most countries, although we are still reminded today
that what topics constitute
the study of mathematics in schools remain to be socio-politically
regulated
by different
government systems. For example, the study of statistics
had been
removed from the secondary mathematics syllabi of schools in some countries.
It is in this climate that numeracy assumes a significant
role in countries such as Australia, New
Zealand, Sweden and United Kingdom. This term is also expressed differently
in different
cultures,
such as 'numerical
literacy',
'functional
mathematics',
and, in the United States,
'quantitative
literacy'.
In Australia,
the notion of numeracy in schools was thrown in the
spotlight
in the 1 999 Adelaide Declaration on National Goals for Schooling in the Twenty-First
Century, in which Goal 2.2 states that students should have "attained the skills of numeracy and
English
literacy;
such that every student should be numerate, able to read, write, spell and
communicate at an appropriate
level" with mathematics as one of eight key learning areas.
Numerous definitions
of numeracy have been made, reflecting
different
approaches
(Perso,
2006), and presenting "an enormous set of conceptual and pedagogical
issues" (Coco, Kostogriz,
Goos, & Jolly, 2006). In this chapter, the approach which appears to have been embraced by the
intended curricula in many education systems (such as those in the UK and the different
states
within Australia)
will be the reference for discussion.
In this 'mathematical
literacy'
approach
(Perso, 2006),
the description
made by the Australian
Association
of Mathematics
Teachers
(1997)
is apt, in which "to be numerate is to use mathematics effectively
to meet the general
demands of life at home, in paid work, and for participation
in community and civic life" (p. 1 5).
Likewise,
Willis
(1998)
associates
numeracy skills with the capability
of one to conduct
' intelligent
practical
mathematical
action in context'.
3.2
Numeracy
in Victorian
primary
schools
In the state of Victoria, Australia,
much of the primary school mathematics curricular activities
has been influenced by the findings and recommendations of two state-level
research projects in
the early 2000s, namely, the Early Numeracy Research Project (Victoria
Department of
Education Employment and Training et al, 2002) and the Middle Years Numeracy Research
Project (Victoria
Department of Education Employment and Training et al, 2001). For example,
students entering the first formal school year (i.e. Grade Prep) will have their individual
numeracy standards assessed (and subsequently
reported) through the School Entry Assessment,
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Development of Mathematics
Education
Teachers 'Final
Report
which takes the form of individualised
interview sessions. While the school subject is formally
called
'mathematics',
the introduction
of initiatives
such as 'numeracy blocks'
in lesson
planning has meant that many primary school teachers in Victoria use the terms 'mathematics'
and 'numeracy' interchangeably,
a phenomenon which is also frequently
observed in the UK.
This is despite the fact that at the intended curriculum level, there is only one 'unforced' use of
the term 'numeracy' in the chapter of the Victorian
Essential
Learning
Standards
for
Mathematics (VCAA, 2005), which spells out the context of the discipline
in the school system
before the outcomes for each learning level are specified
in subsequent chapters. Interestingly,
that one inclusion
may be interpreted
to make an explicit
distinction
between mathematics and
numeracy, where one aim of the curriculum is for students to "demonstrate useful mathematical
and numeracy skills for successful general employment and functioning
in society" (p. 4). The
other occurrences of the term 'numeracy' are in the last section (p. 9), which describes
how
student achievement is reported against the National Numeracy Benchmarks.
While numeracy in the early years of schooling
is mostly (but not exclusively)
concerned with
equipping
students with the necessary understandings
(such as number sense and data sense) to
facilitate
more efficient learning in the middle years and beyond (such as how basic number
sense aids in the development
of algebraic
thinking
as described
in Willis,
2000), the focus of
this chapter is more on the numeracy lessons planned for the middle
years of schooling
(typically,
the last two years of primary school education and the first two years of secondary
school education)
and beyond, where mathematical
tasks in context allow students to apply
what Willis
(1998)
calls the 'intelligent
practical
mathematical
actions'.
In this context,
numeracy appears to be promoted in Victorian primary school classrooms in one of two ways.
In the first way, mathematical
concepts and/or processes are introduced
to students,
before
opportunities
for student applications
and/or investigations
are provided.
In the second way,
numeracy is delivered
interdisciplinarily
and contextualised
in sessions where students work on
investigations
/ projects
to solve identified
or pre-specified
problems,
often in groups. The
necessary mathematical
skills are acquired by students in context, and teachers are expected to
direct student learnings of relevant, emerging mathematical
concepts.
3.3
Does numeracy
mirror
daily
life?
Both these teaching
approaches
are likely to not foster students'
affective
development
in
mathematics
/ numeracy, however. Given the reality of school lessons and the accompanying
constraints,
there will always be a limit to the authenticity
and realism of mathematical
problems in numeracy lessons. This may be especially
so in the first of the teaching approaches,
where the initial
classroom learning of mathematical
concepts and processes may stimulate
students to ask for the reasons why they need to be learnt, or when they will ever be used in life.
However, this does not mean that students experiencing
the second, more holistic
education
approach would feel that what they learn in numeracy lessons are relevant to numeracy demands
in daily life. The validity
of this point is supported by observations
that quite often, authentic
numeracy activities
are not really authentic,
though it must be reminded at the same time that it
is not attributed
to teachers alone.
A few examples may be illustrative.
In Victoria (as in most other educational
systems in the
world), rules for rounding decimal numbers ending with the numeral 5 are not the same as those
used in similar
operations
in the Australian
optical fibre industry
(Fielding,
2003).
Similarly,
Victorian
primary school students may learn to execute rounding to the nearest five cents in
schools, yet children
know that the cash registers
of different retail outlets conduct the same
operation for cash transactions
in different ways (i.e. some round off, some round down, and
others round up to the nearest five cents). The use ofmodals (such as 'might' and 'should')
in
numeracy questions might also stimulate
some children to think that the price for several items
of a particular commercial product needs not be proportional to their unit price (as is the case in
real life),
a situation
which is likely
to be assessed as their teachers
to represent wrong
161
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Cooperation
Project
of Mathematics
Education
computations !
This gap between what constitute school mathematics and daily/workplace
mathematics remains
not only because of the nature of schooling with its constraints
and conditions,
but also because
very often the invisibility
of mathematics
in daily/workplace
tasks has been 'black-boxed'
paradoxically
by the discipline
itself. Latour ( 1 999) refers 'black-boxing'
to
the way scientific
and technical
work is made invisible
by its own success. When a
machine runs efficiently,
when a matter offact is settled, one need focus only on its inputs
and outputs and not on its internal complexity,
(p. 304)
3.4
Emphasising
the
understanding
of others'
mathematics
Thus, introduction
of numeracy in mathematics classrooms (at both the primary and secondary
school levels)
needs to be complemented by an explicit
acknowledgement
by schools and
teachers that the very nature of lesson timetabling
and also often of the intended curriculum
mean that school
mathematics
will neither
be academic
mathematics,
nor will it be
daily/workplace
mathematics.
In education
systems such as those found in Australia
where
numeracy is the aim for mathematics teaching
in schools, teacher explicitness
in highlighting
that not all real-life
situations
can be reasonably
modelled
within the constraints
of lesson
planning is important in sustaining
student interest and faith in the subject, and in emphasising
instead the value in one's capacity to apply mathematical
knowledge in daily life or in the
workplace.
Howmight this capacity be developed?
Williams and Wake (2007) advocate that
to make sense of workplace mathematics, outsiders
need to develop flexible
attitudes
to
the way mathematics
looks, to the way it is 'black-boxed'....
Students
had little
awareness that the College
mathematics
they had learnt was itself idiosyncratically
'conventional',
and that mathematical
conventions might vary in time and place, (p. 338)
The intended
curriculum can play a crucial contributing
role here while still promoting
the
satisfaction
of numeracy outcomes. In fact, it potentially
adds depth to what it means to be
functionally
numerate in the society when students acquire the skill of "making sense of the
mathematics
of others" (Williams
& Wake, 2007, p. 339). These 'others'
would refer to
members of a diversity
of cultures
- national, ethnic, workplace,
religious,
gender, for
examples.
In this regard, Bishop's
(1996)
conception
that cultures universally
engage in six types of
activities
from which mathematical
ideas evolve is a particularly
useful one. At one point or
another in time and space, all cultures find the need to count, locate, measure, explain, design,
or play, and, Bishop (1996)
argued, all cultures perform all these six mathematical
activities.
From the pedagogical
point of view, this is more than enabling
students (and teachers)
to
appreciate
that different
cultures perform mathematics tasks differently,
that, for example, the
Anglo-Australian
parents emphasise
the skill
of counting
in their young, whilst
their
Aboriginal-Australian
counterparts
traditionally
emphasise the importance of their children
to
perceive their respective
locations
within the three-dimensional
space. Without disputing
that
this in itself has rich mathematics
pedagogical
implications,
the focus here is on the potential
for
students to understand mathematical
situations.
Thus, making sense of
'adding up' rather than
sense and understanding.
method to the 'textbook'
understand more of the
constraints,
to explain
how cashiers often provide the change for a cash transaction
through
subtracting
has the potential
of enabling students to go beyond making
It also provides an opportunity
for students to relate this change-giving
method (of using the decimal subtraction
algorithm),
to relate and
operations
of addition and subtraction,
to evaluate relative strengths and
how formal, school and workplace mathematics
support one another.
162
International
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Project
Education
Clearly,
there is also the bonus advantage of fostering
students'
higher-order
cognitive
outcomes.
If a numeracy program seeks to expose students to real-life
mathematics in the daily life or
workplace, then one which identifies
and discusses how mathematical processes are executed by
'others'
can bring the mathematics
out of the 'black-box',
thereby making the underlying
concepts and ideas more visible. It also acknowledges the limitation
of the school mathematics
framework in possibly
discussing
and examining selected mathematical
practices
of selected
cultures,
without
appearing
to advocate that the numeracy developed
through
school
mathematics is sufficient
in equipping
the individual
student to apply related concepts and skills
to the world beyond the school gates. In fact, by highlighting
the cultural
aspects of
mathematical
practices
in numeracy classes, student attainment and mastery of numeracy is
situated within a cultural community of practice, which reflects the very essence of numeracy,
that is, for effective sense-making and negotiation
of demands arising from family, workplace,
community and civis lives.
A numeracy approach which acknowledges the inevitable
gaps amongst academic, school and
daily/workplace
mathematics can highlight
to students that if mathematics is to be regarded as a
form of language for communication, then different
genres of this language evolve from the
different
contexts within which this language is used. Thus, school mathematics is one such
genre, and the extent to which policy makers desire to inculcate
numeracy in the young
generation through school mathematics will be reflected
in its closeness
with the genres of
daily/workplace
mathematics. Most importantly,
such an acknowledgement
can address student
sceptism abut the value of school mathematics,
and the explicitness
can foster student
appreciation
of the genre of school mathematics.
Students' cognitive
'movements' within and across the different
genres of mathematics is also
educationally
valuable
in that a meaningful
appreciation
and understanding
of how 'others' do
mathematics can promote positive
development of their individual
cultural
intelligence
[CQ].
First conceptualised
by Earley and Mosakowski (2004),
CQ extends the notion of emotional
intelligence
and is a measure of the extent to which an individual
who is
an outsider
unfamiliar
colleagues
[to a particular
culture]
has a seemingly natural ability to interpret someone's
and ambiguous gestures in just the way that person's compatriots
and
would, even to mirror them. (p. 139)
A numeracy curriculum which provides students opportunities
to relate mathematical
concepts
and ideas to its various genres in different
occupational,
ethnic, gender or other categories
of
cultures is an exercise in interpreting
these other cultures' unfamiliar
ways of doing mathematics.
This hidden curriculum of cultivating
students'
individual
cultural
intelligence
extends beyond
mathematics/numeracy
learning, to the learning of other educationally
desirable
skills of social
learning, such as interpersonal
skills.
3.5
Closing
remarks
The discussion
above re-examines the role of the cultural aspects of mathematics in the teaching
of numeracy in schools. It is expected to be relevant to educational
systems in both developed
nations and developing
ones that seek to benefit from being part of the knowledge-production
economies. Participation
in these economies (as compared to agricultural,
manufacturing,
service and information economies) require more than ever before a work force whose members
are functionally,
if not critically,
literate
and numerate. The argument made here is that
achieving
meaningful
and sustained
numeracy rates amongst a country's students may be
supported by intended and implemented
curricula which acknowledge the differences
amongst
academic, school, daily-life
and workplace mathematical
practices.
By promoting
student
understanding
of culturally-different
ways of doing mathematics,
the black-boxing
of
163
International
Cooperation
Project
Education
mathematical
concepts and ideas in daily life and in the workplace can be reduced. This should
optimise
student progression
through the school years with an increasing
understanding
and
confidence in relating academic and school mathematics to the demands of tasks they encounter
or will encounter in the respective
personal and working lives they lead, with the additional
potential
of cultivating
their individual
cultural intelligence
in the process.
Reference
Australian
Association
of Mathematics
Teachers. (1997).
Numeracy = Everyone's business.
Adelaide,
Australia:
Australia
Department
of Employment,
Education,
Training
and
Youth Affairs.
Bishop, A. J. (1996,
June 3-7). How should mathematics teaching in modern societies relate to
cultural
values - somepreliminary
questions.
Paper presented
at the Seventh
Southeast Asian Conference on Mathematics
Education, Hanoi, Vietnam.
Coco, A., Kostogriz,
A., Goos, M., & Jolly,
L. (2006).
Dangerous
liaisons:
Home-business-school
numeracy networks. Paper presented
at the Australian
Association
for Research in Education
2005 Conference, Parramatta, Australia.
Earley, C. P., & Mosakowski, E. (2004).
Cultural
intelligence.
Harvard Business Review, 82( 1 0),
139-146.
Fielding,
P. (2003).
Are we teaching
appropriate
rounding
rules? In B. Clarke, A. Bishop, R.
Cameron, H. Forgasz & W. T. Seah (Eds.),
Making mathematicians
(pp. 395-401).
Victoria,
Australia:
The Mathematical
Association
of Victoria.
Latour, B. (1999).
Pandora 's hope: Essays on the reality of science studies. Cambridge,
MA;
London: Harvard Univeristy
Press.
Perso, T. (2006).
Teachers of mathematics or numeracy? Australian
Mathematics Teacher, 62(2),
36-40.
Victorian
Curriculum
and Assessment Authority.
(2005).
Discipline-based
learning strand:
Mathematics.
East Melbourne,
Victoria:
Victorian
Curriculum
and Assessment
Authority.
Victoria Department of Education Employment and Training, Catholic
Education
Commission
of Victoria,
& Association
of Independent
Schools
of Victoria.
(2001).
Successful
interventions:
Middle years numeracy research project 5-9 (stage 2) (Final
report).
Melbourne.
Victoria
Department of Education Employment and Training,
Catholic
Education
Commission
of Victoria,
& Association
of Independent
Schools of Victoria. (2002).
Early Numeracy
Research Project (1999-2001):
Summary of the final report (Report).
Melbourne:
Victoria
Department of Education and Training.
Williams,
J., & Wake, G. (2007).
Black boxes in workplace mathematics. Educational Studies in
Mathematics, 64, 3 17-343.
Willis,
S. (1 998, September 27-30). Numeracy for the(ir) future: Rite or right? Paper presented
at the National Conference of the Australian
College of Education, Canberra, Australia.
Willis,
S. (2000).
Strengthening
numeracy: Reducing risk. In Improving numeracy learning (pp.
3 1 -33). Melbourne, Australia:
Australian
Council for Educational
Research.
164
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4.
Cooperation
Minority
4.1
Students
: Reviewing
Hideki Maruyama
National Institute
Project
and Teacher's
International
for Educational
Development
Education
Teachers 'Final
Report
Support
Assessment
Policy
of Mathematics
Results
and Cultural
Approach
Research (NIER)
Introduction
One of the most influential
international
comparative studies on education is the Programme for
International
Student Assessment (PISA) for recent years. The program has been started as a
response to the need for cross-nationally
comparable evidence on student performance by the
Organisation
for Economic Co-operation
and Development (OECD) in 1997, and the first
assessment was conducted in 2000. The results of PISA revealed wide differences
in the extent
to which countries succeed in enabling young adults to access, manage, integrate, evaluate and
reflect on written information
in order to develop their potential.
At the same time, the results
showed the commondifficulties
across countries and unique issues within them. Some countries
recognize the significant
variation of the performance level among their children
and concerns
about equity in the distribution
of learning opportunities,
although they have the equivalent
of
several years of schooling
and sometimes despite high investments in education as the other
better performed countries.
Some European countries found reasons why the disparity
exists between student's high and
low performances. Immigrant students have a lot of different backgrounds from natives as both
personal concerns like a language and social issues such as economic conditions. The results of
the international
assessments
introduce the clear difference,
as quantitative
data, and the
governments have started to focus on students' socio-economic backgrounds for better education
system. And meanwhile, teachers in a host country for the immigrant students are natives in
many countries.
Our IDEC research project started to describe
phenomena of mathematics
education and
classroom interaction
in the wide range of countries. We are prone to imagine that instructions
in classroom cause lower performance of the students in the developing
countries because of its
poor pedagogical
approach such as chalk and talk. The results of the first and second year
survey can be interpreted
as a finding that the students who do not use a test language as their
first language achieve lower academic performance. The author believes the language problem
is an explicit
aspect of cultures which students generally
have in their classrooms because by
and large they are always in multi-cultural
environment and the interaction
between a teacher
and students must be complicated
like those in European classrooms. Minority students need
more supports especially
from teachers because the teacher's
role in classroom is rather a
controller
in the project
countries
and their outer views are limitedly
sensitive
to the
backgrounds of the minority students.
In this review, the author summarizes the PISA results of immigrant students in mainly Europe
with literature
and analyzes the backgrounds
of the immigrants
from the cultural point of view.
The author also tries to discuss the context of the classroom in developing
countries in which
local teachers might miss the cultural backgrounds of the minority students.
4.2
Findings
from the Assessments
Concerning
the Immigrant
Students
Many types of education
assessments are available
when we try to analyze student's
performance today. Here we look at PISA and its related research because it is one of the most
influential
to education in the world today. Following the brief description
of the world wide
assessment, let us see the performance of immigrants and their language problem.
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Assessment
PISA seeks to measure how the 1 5-year-old children,
as the age of ending compulsory schooling
in many countries, are prepared to use their knowledge and skills to meet real life challenges
of
today's knowledge society, rather than merely on the extent to which they have mastered a
specific school curriculum. PISA is the most comprehensive and rigorous international
program
today for the participating
countries/regions
to assess their student performance and to collect
data on student, family, and institutional
factors that can help to explain
differences
in
performance.
PISA has three cycles of its implementation
procedure
and three domains of student's
knowledge and skills
called "literacy,"(1)
namely, reading, mathematical,
and scientific
ones.
Each cycle covers all the domains but its focus is mainly on one of the three domains. The first
PISA assessment was conducted, focusing on reading literacy,
in 2000 and on mathematical
literacy
in 2003. More than a quarter of a million
students
and their school managers
participated
in the each. There will be the first final cycle for about 60 countries,
including
30
OECD countries, on the scientific
literacy in 2006. Besides the literacy,
the participants
also
responded their family and economic backgrounds, their motivation
to learn, their beliefs about
themselves, and their learning strategies.
4.2.2
Lower Performance
of lmmigrant(2)
Students
in the Assessments
The results of the series of PISA provide us with the data the students whose parents are not
native show lower performance than native students in many countries. The greatest gap, of 93
points in mathematics scores, is in Germany, and students themselves born outside the country
tend to lag even further behind, in Belgium by 109 points.(3)
While circumstances
of different
immigrant groups vary greatly, and some are disadvantaged
by linguistic
or socio-economic
background as well as their migrant status itself, two particular findings are worrying for some
countries. One is the relatively
poor performance even among students who have grown up in
the country and gone to school there. The other is that after controlling
for the socio-economic
background and language spoken at home, a substantial
performance gap between immigrant
students and others remains in many countries:
it is above half a proficiency
level in Belgium,
Germany, the Netherlands,
Sweden and Switzerland.(4)
Schnepf (2004)
used the results of ten countries(5)
which had participated
in IEA's TIMSS and
PIRLS in addition
to PISA because all the three assessments collected
information
in the same
format regarding immigration
variables
of their parents origin and language used at home.(6) She
explained
the immigrants
achieved significantly
lower test scores than natives in almost all
countries and surveys. She concludes the promoting language education of immigrant students
and decreasing
school
segregation
are likely
to have a positive
outcome on students'
performance, based on the three findings from her statistical
analysis:
1) the immigrants'
socio-economic
background
might be lower than natives and be immigrants'
educational
disadvantages
in some countries;
2) immigrants'
educational
disadvantage
might derive from
their problems of integration
into the host country; and 3) the process of selection/acceptation
of
immigrants to the host country is likely to impact upon their achievement results.
Ammermuller (2005)
used the data from PISA extension study(7) in Germany and tried to find
determinants
that the results of the immigrant students were lower than native students. He
found that the use of different
language
at home from German in classroom is typical
characteristics
for the immigrant
students,
in addition
to the higher grade level of German
students and more home resources as measured by the amount of books at home. These factors
explained
the test score gap would decrease by 40 to 49 %, if the immigrant students had had
the same returns to student background
as native students.
However, he also found that
differences
in parental education and family situation were less influential.
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Rangvid (2005)
made an analysis of the immigrant students' performance in Denmark by using
the results
of PISA-Copenhagen
which is based on international
PISA study design. The
PISA-Copenhagen
results suggest similar performance between the native students and the
students who have one immigrant parent, but the mean scores of the immigrant students who
have both non-native parents are much lower. She also pointed out the immigrant students
experienced
lower teacher expectations
and the peer composition
at schools attended by
immigrant students was potentially
less conducive to academic achievement.
The results of the international
assessments show that immigrant students achieve lower level of
knowledge and skills
than natives in many European countries.
We can assume that the
language proficiency
of the immigrant
students
is one of the main reasons why their
performance is lower than natives.
4.2.3
Teaching
Limited
English
Proficient
Students
Secada and Carey (1990)
argue school practices might forge the links that the level of English
language proficiency
affects mathematics achievement. They believe that children enter school
with a broad range of understandings
about mathematical
concepts. Instruction
should build
upon and develop the children's
knowledge so that it would provide children's
learning
in
mathematics and contribute to their confidence about their abilities.
They describe one effort to
accomplish this approach called Cognitively
Guided Instruction
(CGI).(8)
Although
the CGI assessment focuses on the processes by which students get an answer and
seems to be quite learner-centered
approach to some countries in the IDEC project, the unique
procedure relates with intercultural
communication. To start assessment of student's thinking,
a
teacher should provide students with counters, pose a problem, and if the student responds,
follow up by asking how he/she figured it out. The teacher then uses additional
questions to help
him/her clarify what it meant. Alternatively,
if a student does not solve the problem correctly,
the teacher has at least five options.(9) The point is to begin a conversation about the problem.
Many limited English proficient
students receive little encouragement to speak about their ideas
and many girls are socialized
to defer to boys. If teachers tend to call on students who answer
first, the rest will often be left out of the conversation. Therefore, teachers need to reach out to
these children and to be sure that they are included in the classroom's processes.
They recommend that students should be
during mathematics.
As a result, students
the teacher. This validates
their thinking
students are a source of knowledge in the
4.3
able to discuss how they are thinking
about problems
learn mathematics from one another as well as from
and students begin to recognize that both teacher and
classroom. They might share the backgrounds more.
Education
for the Cultural
Minority
: integration
as policy and personal
level
We have seen the lower performance of the immigrant
students. In addition
to the language
issue, minority
student's
cultural
backgrounds
call for our attention
to more influence
in
classroom interaction
and their academic performance than language. European countries such
as Germany, Denmark, and the Netherlands
have received immigrants
in past decades have
emphasized the efforts for integration
policy of the immigrants
into the native societies
as a
solution.
But the policymakers
found the results of the assessments show the integration
policy
stil
faces challenges.
At school level, many teachers understand
immigrant students'
poor
achievement stems from their language problem and different cultural backgrounds.
Immigrants
also have difficulties,
however sometimes impossibility,
to cope with different
environments. Immigrant students live with host culture at school and original
culture at home.
Having focused on language, when their language ability is not enough to understand instruction
in class, they not only feel less confident in learning lessons and participating
classroom
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activities
but also hesitate
to ask what they do not understand and develop little concept of
mathematics at last.
This section deals with cultural experience of the immigrant students, their cultural learning
process, and possible
supports by teachers. Descriptions
can apply to some examples of IDEC
project because this paper takes the problems of the immigrant students in mostly European
countries as similar phenomena to the "cultural
minority"
groups.(10)
Before proceeding,
we
should also remind ourselves of the risk of overgeneralization.
4.3.1
Cultural
Experience
of Immigrants
Immigrant students live in two or more different cultures(ll)
and are also in co-culture groups.(I2)
Parents' views and values affect students, especially
in childhood
prior to schooling
so strongly
that children
have bias and feel uncomfortable
when they first face different views from other
students
in school. For example, they experience
the different
manners of expression in
low-context(13)
culture in Europe and feel the large gap between their own in-group and other
out-group cultures.{14)
And meanwhile, the native students of dominant culture, or majority
students, also feel difference
of paralanguage
such as gestures and facial expression
when
interacting
with "strangers."
As mental defense mechanism that they try to avoid the
unfamiliarity
and reduce uncertainty,
they tend to stay with those who have similar backgrounds
for their comfort through the interaction
among students.
This grouping process is different
from the psychological
stage as gang age but rather based on
mental state called ethnocentrism.
Ethnocentrism
is a universal tendency of the belief that one's
own group or culture is superior to other groups or cultures. People generally
interpret
other
cultures according to the values of their own. When the students consciously
separate in- and
out-groups, they attribute
the other group's mistake, for instance, to the characteristics
of the
whole group but own group's to personal. A German female student sees a Turkish student
wearing headscarf in physical
education class, she tend to thinks all the Turkish girls wear it.
But she does not think that all the German girls go to Sunday school because it depends on
family's attitude
to daughter. This is not a serious problem very much until students start to
develop
prejudice
from the bias after cultural
encounter because they are barrier
to
understanding
others. The students need a mentor to follow up their observation
and
understanding
the difference properly.
Both minority and majority groups generally have bias and possibilities
of conflicts between the
two. The host society is basically
more affirmative and supportive
to the majority,
and the
minority groups have to follow many of its social norms. The society enforces the minority
integration
or assimilation
to be the citizen so that a strong reaction such as marginalization
sometimes comes up from the minority group. It is understandable
that a group of immigrant
students seems culturally
homogeneous to the majority students and native teachers because
their appearance and attitude resemble each other. And their communication can be categorized
as intra-cultural
communication
if we compare communication between explicitly
different
cultures. The large scale survey and analysis also categorize they are the same group; otherwise
there are too many cultural-specific
cases. When we easily categorize
and judge them based on
ethnocentric
view, prejudice
overcasts our eyes.
Here, we need to shift our views. What we think as intra-cultural
communication can be actually
intercultural
when we see their experiences by emic approach. This view is originally
from the
standpoints
of Pike (1967)
who has generalized
from phonetics and phonemics to what he has
called the "etics" and "emics" of all socially
meaningful
human behavior. Whatever the names
one may choose to call them, the concepts of etic and emic are indispensable
for understanding
the problems
of description
and comparison.
Etic is pertaining
to a concept that is
culture-general
and is therefore easy to describe when we examine cultures from outside. Emic
is pertaining
to a concept that is culture-specific
and is therefore difficult to translate from one
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language to another when we examine a culture
emic is internal
view.(15)
Development
of Mathematics
from the inside.
It might be much easier for us to perceive the various
cultural
minority
if we try to observe from the inside.(16)
the biggest
and typical
issues which majority/minority
classroom. Parker-Jenkins
(1 995) illustrates
an example:
In short,
Education
etic is external
and
backgrounds
of immigrant
students
or
Religious
consciousness
may be one of
students
and teachers
concern in
"Muslim children are taught to respect and not question elders or those in authority
and accordingly
they may appear to be stereotypically
passive and accepting. If a child does not question teachers it
does not automatically
denote lack of interest or intelligence
but rather respect for their authority and
position.
Furthermore, the value placed on deference to authority,
and haya or modesty within the
Islamic consciousness,
helps to explain the difficulties
and contradictions
some Muslim children
may experience in participating
in [British]
state school activities
which call for assertiveness
and
extrovert behaviour."'
'
4.3.2
Acculturation:
Integration
orAssimilation
of the Immigrants
The immigrant students sometimes have no choice but follow
students learn appropriate
behaviors
by living in a certain
acculturation
which could contain four types of strategies.
the dominant group's norms. The
cultural
context. This is called
Herskovits
and other American anthropologists
(1936)
proposed the following
definition
of
acculturation.
"Acculturation
comprehends those phenomena which result when groups of
individuals
having different
cultures come into continuous first-hand contact, with subsequent
changes in the original
cultural patterns of either or both groups."(18) Acculturation
differs from
cultural change in which the source of change is internal
within the culture and either from
enculturation.(
19)
Berry ( 1 992) categorizes
acculturation
strategies
in which individual
or group wishes to relate to
the dominant
society
into
four patterns:
integration,
assimilation,
separation,
and
marginalization.
They are conceptually
the result of an interaction
between ideas deriving
from
the culture change and the intergroup
relations.
In the former the central issue is the degree to
which one wishes to remain culturally
as one has been as opposed to giving it all up to become
part of a larger society;
in the latter it is the extent to which one wishes to have day-to-day
interactions
with members of other groups in the larger society as opposed to turning away from
other groups and relating only to those ofone's own group.
When these two central issues are posed simultaneously,
a conceptual
framework [Fig. 1] is
generated that posits four varieties of acculturation.
When an acculturating
individual
does not wish
to maintain culture and identify
and seeks daily interaction
with the dominant society, then the
assimilation
path or strategy is defined. In contrast, when there is a value placed on holding onto
one's original culture and a wish to avoid interaction
with others, then the separation alternative
is
defined. When there is an interest in both maintaining
one's original culture and in daily interactions
with others, integration
is the option; here there is some degree of cultural integrity
maintained,
while moving to participate
as an integral part of the larger social network. Integration
is the strategy
that attempts to "make the best of both worlds." Finally, when there is little possibility
or interest in
cultural maintenance, and then marginalization
is defined.(20)
Although the officials try to promote integration
of immigrants into the society, the immigrants
sense its uncertainty and meanwhile the host local people are not interested
in the promotion
very much. Parker-Jenkins
(1995)
draws multiculturalism
in Britain
and details
a phase of
assimilation
took place along with great restriction
of immigration
to the country in 1950s.
The author also interviewed
German scholars about immigrants'
integration
in Germany and
recognized
some cases could
be categorized
as assimilation
which
possibly
leads
marginalization.(2I)
The immigrants
or minority
group of people follow the norms of the
majority society with mental conflict.
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Issue 1
Is it considered
to be of value to
maintain cultural
identify and
ch aracteristics?
"YES"
"NO"
Issue 2
Is it considered
to be of value
to maintain relationships
with other groups?
Fig.
4.3.3
Support
from Teacher
I
I
"YES"
Integration
Assimilation
"NO"
Separation
Marginalization
1 Four varieties
of acculturation
(Berry,
1992)
in Classroom
The minority students need supports from teachers and other students. Friends are the strongest
supporter in and out of classroom but if no one wants to be a friend, then the immigrant students
have only teachers to ask for help. Teachers become more important being today not only for the
academic achievement of minority students but also for their coping strategy with the dominant
culture. OECD recently
operated a project of policy
research
on teachers in its member
countries,(22)
and its report introduces the importance of teachers:
The demands on schools and teachers are becoming more complex. Society now expects schools to
deal effectively
with different languages and student backgrounds, to be sensitive to culture and
gender issues, to promote tolerance and social cohesion, to respond effectively
to disadvantaged
students and students with learning or behavioural problems, to use new technologies,
and to keep
pace with rapidly developing fields of knowledge and approaches to student assessment. Teachers
need to be capable of preparing students for a society and an economy in which they will be
expected to be self-directed
learners, able and motivated to keep learning over a lifetime.(23)
Teachers are not allowed to give lessons only for subjects anymore but need to keep learning in
today's knowledge-based
society. Communication
is clearly
important
in classroom among
students, and teachers actively receive feedbacks from them. In addition
to receiving a message
from them, teachers may have to imagine students'
thoughts based on understanding
on their
backgrounds. It requires more efforts than before so that teachers might lose what to do first.
To imagine the student's
needs, the first and explicit
aspect of their backgrounds
is their
language. Their language proficiency
and teacher's
understanding
of their language at home
help both students and the teacher. The students' explanations
may be tentative
not only because
of content mastery but also because of the language used in classroom.'24^
Moreover, even when
using their native languages, many students have difficulty
in expressing
their thoughts
in as
sophisticated
a manner as teachers might like. Teachers must recognize students judge people
who sound like they know what they are talking
about have knowledge,
while those who
express themselves
poorly do not have such knowledge.
Teachers need to begin with what
students
understand
and with how they can express their
understandings.
Therefore,
mathematics teachers are required the knowledge of both subject and cultures today.(25)
To know more about cultures,
Hofstede's
framework can be helpful.
Geert Hofstede (1984)
categorizes
four dimensions
of values by surveying across 50 countries(26):
1) power distance, 2)
uncertainty
avoidance,
3) individualism
and collectivism,
and 4) masculinity
and femininity.
Power distance is a value regarding
the degree to which a society accepts the fact that power in
institutions
is distributed
unequally.
If it is small,
equality
is highly
sensed. Uncertainty
avoidance
is the attempt to understand
the other person in a communication
setting
by
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Development
of Mathematics
increasing
predictability;
brought about by the high need to understand
interpersonal
situations.
If a society has strong uncertainty
reduction,
deviant behaviors
of others. Individualism
gives more importance to
group, and collectivism
is vice versa. Masculinity
refers to emphasis
achievement
and etc., while femininity
prioritizes
compassion to the
quality
of life and etc. Hofstede compares working staff at country
classroom can identify
themselves with the framework and locate the
behaviors within culture and between co-cultures. This shift of viewpoint
from the teacher's etic views.
Education
Teachers 'Final
Report
oneself and others in
it is less tolerance
to
individual's
will than to
on heroism, argument,
weak, human relation,
level, but teachers in
background of student's
can be emic approach
When teaching side shifts its view to the inside, teachers would be more able to understand
minority student's backgrounds besides language. As an example of Muslim students in Britain,
Parker-Jenkins
(1995)
categorizes
student needs into four:(27)
1)
As religious/cultural
needs: Time of school assemblies
should be coordinated
for daily
prayers. School diet should be considered
and understand the need of fasting.
Dress
code should be flexible,
especially
in physical
education for girls, etc.
2)
As curriculum
needs: Sex education should be paid more attention, especially
need for
HIV issue. Language instruction
for minor languages should be encouraged. Islamic
dimensions
should be promoted, etc.
3)
As linguistic
4)
As general needs: Home-school links should be emphasized
because enlisting
parental
support is vital to all attempts to accommodate Muslim needs within the school system.
Muslim organizations
are encouraged to become more proactive in attempting
to bring
about change within the educational
system for school governance.
needs:
Support
for English
language
competence/acquisition,
etc.
She conducted the survey on the above needs and practices of British
schools and found the
similarities
and differences
between the perceptions
of head teachers in state schools and private
Muslim schools over Muslim children's
needs. The similarities
were in the issue of religion
and
religious
observance,
and the differences
were the "conceptualization
of religion"
in the
children's
lives,
English
language
acquisition,
effective
home-school
links,
and balanced
curriculum.
She points out that English
language acquisition
is considered
the major area of
concern by state school administrators
because it is the medium of instruction
in British
schools
but opportunity
for various groups to learn their own language need to be made for its cultural
importance
and the matter of parity of languages.
She also pointed that curriculum
as a
"transmission
of the culture" transmits
minority
cultures
and the right balance between the
indigenous
and minority
cultures
is an area of concern. She suggested
understanding
the
processes of assimilation
and acculturation
within multicultural
education
is pertinent
through
learning
from each other and emphasized
that teachers
cannot be expected to translate
educational
theory into practice without adequate training
and the selection
and recruitment of
teachers from minority backgrounds
are needed.
4.4
Conclusion:
Cultural
Sensitivities
of Teachers
In this review, we have seen the immigrants'
low performance shown by the international
assessments and cultural backgrounds,
their cultural
experiences
and personal coping strategies,
and teacher's
room to learn to shift the view to the insider.
We may summarize them in the
following
three.
4.4.1
Importance
of teacher's
role
The largest source of variation
in student learning
is attributable
to differences
of students'
personal
and cultural
backgrounds
such as their individual
abilities,
attitudes,
family,
and
community. These factors are difficult
for the outsiders
to influence.
On the other hand, factors
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Teachers ' Final
Report
concerning
teachers and teaching
are the most important
influences
on student learning.
Teacher's education and experience influence
on student achievement.
But as OECD reports "a
point of agreement among the various studies is that there are many important aspects of teacher
quality that are not captured by the commonly used indicators
such as qualifications,
experience
and tests of academic ability,"(28)
we can focus more on teacher's potential
to recognize the
student backgrounds
for their learning.
In the IDEC project,
we shall also remember more
cultural-specific
aspects, for example, the traditional
African education(29)
is based on oral
communication as Reagan (2005)
examines. The traditional
education
and training
have its own
contexts, and local teachers must be tolerance and sensitive
to them. This is because "formal
schooling
played
important
roles in many non-Western traditions
as one of fundamental
distinctions."(30)
4.4.2
Cultural
sensitivity
OECD (2005)
introduces
the five broad areas for the development
of professional
knowledge
and expertise in teaching, as Shulman (1992)
identified'3!):
teacher's behavior, cognition,
content,
character,
and knowledge of and sensitivity
to cultural,
social, and political
contexts and
environments of the students. The fifth area treats teacher's sensitivity
to cultural
backgrounds
of the minority
students and it is what this paper finds the importance. Not only OECD but all
teachers who have minority
students already know classrooms are becoming increasingly
diverse with students of different cultural and religious
backgrounds.
"Teachers are expected to
work for social
cohesion
and integration
by using appropriate
classroom
management
techniques
and applying
cultural knowledge about different
groups of students."(32)
Teachers are symbolized
powerful authority
in some cultural contexts. For this reason, teachers
themselves
are required for tolerance and sensitivity.
Children
with limited English
proficiency
generally
have very difficult time learning in regular classrooms and easily lag behind leading to
lack of confidence
in their academic
performance.
Less confidence
in language
and
communication
negatively
affect student's
motivation
to learn, and a teacher sees student's
attitude
and has low expectation
so that the vicious cycle of impotence appears within the
minority students in classroom. In the similar context, special attention
should be paid to girls
because they are not encouraged to express their thoughts more than boys and their performance
is lower than male as shown by the international
assessments. This relates with the concepts of
empowerment of women and equal opportunity to access the basic education
in the field of
international
development
cooperation.
Teachers have to be the all sensor in classroom in any
cultures.
4.4.3
Inclusive
atmosphere
in classroom
The role of teachers becomes more important today, and they can create the cooperative
atmosphere in classroom to promote more interaction
among students. This is not only the cases
in Europe but should also be African cases. Dei (2005)
introduces
the case of education policy
in Ghana and emphasizes the inclusive
education,
even though Ghana has the goal of national
integration,
because postcolonial
education in Ghana denies heterogeneity
in local populations.
He explains no attempts have been made to explore majority-minority
relations
in schooling
or
to draw comparisons with approaches to minority education in other pluralistic
contexts. This
situation
missed the needs of students from minority
backgrounds,
and people take the
difference
negatively.
For example, "girls are denied access to schooling
precisely
because of
this marker of 'difference'
that it they are different from boys, or more accurately (in terms of
power, privilege
and dominance),
they are not boys."(33) He also introduces
an interview result
as reaction to the minority
student:
"For minority
students it is very important to see that
everybody can equally perform regardless
of background.
Specially
here some people tend to
look down on them, and you know, it hurts them psychologically.
So, if in all places some of
their community members serve as role models, it would really help them a lot."(34) When we
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observe a classroom in Ghana, we are hardly
difference
which the local people recognize.
Development
able,
ofMathematics
from the outside
Education
Teachers 'Final
views,
Report
to understand
the
Notes:
(1) The definitions
of the each literacy
are as follows
(OECD,
2003):
•E Mathematical
literacy:
an individual's
capacity
to identify
and understand
the role that
mathematics
plays in the world, to make well-founded
judgments
and to use and engage
with mathematics
in ways that meet the needs of that individual's
life as a constructive,
concerned and reflective
citizen.
•E Reading literacy,
goals, to develop
understanding,
using and reflecting
written texts, in order to achieve
one's knowledge and potential
and to participate
in society.
one's
•E Scientific
literacy:
the capacity
to use scientific
knowledge,
to identify
questions
and to
draw evidence-based
conclusions
in order to understand and help make decisions
about the
natural world and the changes made to it through human activity.
(2) There are various definitions
of immigrants have been employed in the literature.
Some take
students born to one immigrant and one native parent as immigrants (Ammermuller
2005),
other label only students with two immigrant parents as immigrants (OECD, 2001). As we
are seeing the immigrant students' performance of PISA here, the immigrant students should
be school children
whose parents are not natives of the host country.
(3) OECD, 2003,
p. 172
(4) See OECD, 2003,
Table 4.2h, for more detail.
(5) Australia,
Canada, France, Germany, the Netherlands,
the UK, and the USA.
New Zealand,
Sweden, Switzerland,
(6) Her explanation
shows the difference
between PISA and TIMSS and the language influence
to mathematics
achievement:
"We might expect that immigrants'
educational
disadvantage
is
smaller in subjects where language skills are generally
ofa lower importance like in math.
However, achievement in math in PISA is measured by applying
the 'life-skill'
approach
related to open-ended questions on wordy descriptions
of 'real life' situations.
Hence, it is
not necessarily
surprising,
that immigrants
do not fare better in math in PISA. This result
stands in contrast to TIMSS: for all countries
immigrants'
educational
achievement gap is
significantly
lower in TIMSS math than in TIMSS science. Hence, there seems to be the
tendency that immigrants'
educational
disadvantage
is smaller in technical
subjects
as long
as achievement is assessed in a more curriculum based approach by using predominantly
multiple-choice
questions
(Schnepf,
2004: 13)."
(7) It is so-called
PISA-E in Germany which includes over 34,000 students compared to 5,500
students in the German OECD-PISA dataset. PISA-E data are detailed
data on student
performance for Germany and show 8 1% of the students have both German parents and 19%
have one or both non-German parent. The data also show 40% of the 19% students speak
another language than German at home.
(8)
CGI is based on the following
four interlocking
assumptions:
1) Teachers should know how
specific mathematical
content is organized
in children's
minds; 2) Teachers should make
mathematical
problem solving the focus of their instruction
of that content; 3) Teachers
should find out what their students are thinking
about the content in question;
and 4)
Teachers should make instructional
decisions
based on their own knowledge of their
students' thinking.
173
International
Cooperation
Project
(9) First, pose a similar problem. If the student
simplify
the language of the problem even
context to a similar problem. The contexts
home cultures and backgrounds.
Forth, the
native languages. If all else fails, the teacher
will be surely able to solve.
Development
of Mathematics
Education
Teachers ' Final
Report
needs more help, secondly, the teacher might
further. The third option is to add localized
added to story problem is to refer to the children's
teacher translates
the problem into the students'
might try an easier problem which the students
(1 0) The minority groups here should be those who have different
cultural backgrounds
in
classroom. This includes
the students who are not fluent in a test language and who can
hardly understand
local contexts in classroom. As the results of the first year IDEC survey
in Ghana show English is a key competency for the students because the teachers reported
the difficulty
for the students were sometimes a very language problem. Even the Chinese
case gives us a variety of the students' background for languages. As the author explained
above, the language is one of explicit
aspects of culture as a symbol of power and personal
ability.
(1 1) Culture
is defined in many ways in literature.
deposit of knowledge, experience, beliefs,
timing, roles, spatial relations,
concepts of
possessions
acquired by a group of people
and group striving
(Samovar, L. and Porter,
Here we rather take it generally
as "the
values, attitudes,
meanings, hierarchies,
religion,
the universe, and material objects and
in the course of generations
through individual
R., 1991)."
(1 2) co-culture:
in another
a dominant
a group of people
culture.
living
within
culture,
yet having
dual membership
(13)
A form of communication in which the explicit
coded message contains almost all of the
information
to be shared. In contrast, little information
is spoken out in a high-context
culture. The four differences
between high- and low-context are: 1) verbal message is
importance and shared in low-context so that people need to observe the situation
to gain
the information;
2) people in high-context
trust oral expression
less than those in
low-context; 3) people in high-context
are more sensitive to non-verbal communication and
are more reading the situation;
and 4) people in high-context
think anyone can achieve
communication without words so that do not use words as much as those in low-context.
(14)
in-group:
the ground that one belongs
who are not members of an individual's
to (relatives,
group.
clans,
organizations);
out-group:
people
(1 5) Pike wrote more about etic/emic:
creation / discovery
of a system; external / internal view;
external / internal
plan; absolute / relative criteria;
non-integration
/ integration;
sameness /
difference
as measured vs. systemic;
partial / total data; preliminary
/ final presentation;
and etc. In addition,
etic approach is different
from structurism
which compares cultures
and leads universality
between structures in which events cause and effects. Researchers
study cultures with the imposed etic but need to make it from emic. In other words, we
need to pause ourjudgment
as observing a different culture with emic approach and switch
to derived etic as describing
the culture in our language.
(16)
However, the author admits his criteria for taking anthropological
approach are not constant in this paper for the moment.
(17)
Parker-Jenkins,
(18)
Red field,
Linton,
1995,
or psychological
p. 28
and Herskovits,
1936,
p. 149
(1 9) Enculturation
refers to the process by which the individual
learns the culture of the
belonging
group. Socialization
is also used for the same meaning. However, Herskovits
(1 948) defined enculturation
as "the aspects of the learning experience which mark off man
from other creatures, and by means of which, initially,
and in later life, he achieves
174
International
Cooperation
Project
Development
of Mathematics
competence in his culture" and socialization
as "the process
individual
is integrated
into his society (p.38-39)."
1992,
Education
Teachers 'Final
Report
by means of which an
(20)
Berry,
p. 278-279
(21)
Interviews
were conducted as semi-structured
and informal way at the different
time and
venues as the author had opportunities
to communicate with German scholars. The
interviewees
were: Ms. Inge Steinstra|3er,
Deputy Director, VHS in Bonn (Dec. 2, 2004);
Dr. Heribert
Hinzen and Dr. Werner Hutterer, IIZ/DVV (Dec. 19, 2005);
and Dr. Raphaera
Henze, Ministry
of Science and Health of Hamburg (Jan. 1 1, 2006).
(22)
"Attracting,
Development and Retaining
over the 2002 to 2004 period.
(23)
OECD, 2005,
(24)
In the studies we reviewed above, the results explain the language as "one aspect of the
integration
issue is the pupils'
capacity
to communicate in the language of the host country.
Regression
analysis showed consistently
across surveys that speaking
a foreign language at
home decreases pupils'
achievement
greatly in all countries
compared. (Schnepf,
2004,
p.36)"
(25)
Secada & Carey say "the instruction
for students from culturally
diverse backgrounds
often
does not take account of the everyday sources of their informal knowledge,
i.e., the
knowledge that they bring from home.... teachers ofLEP [limited
English
proficient]
students not only need to focus on their students'
informal understandings,
but they also
need to seek those understandings
in ways and in settings that are other than commonly
supposed."
(26)
Later, more than 80 countries,
Hofstede, 2005).
(27)
Parker-Jenkins,
op. cit. p. 1 18-120.
(28)
OECD, op. cit,
p. 27
(29)
Because scholars have tended to equate "education" with "schooling,"
and because they
have consistently
focused on the role of literacy and a literary tradition,
many important
and interesting
- indeed, fascinating
- traditions
have been seen as falling outside of the
parameters of "legitimate"
study in the history and philosophy
of education (Reagan, 2005,
Effective
Teachers"
OECD Project
conducted
p. 97
including
Arabic
and African
countries
(Hofstede
and
p.6).
(30)
Reagan,
2005,
p. 248
(3 1) "Shulman,
L., 1991, Ways of seeing, ways of knowing, ways of teaching,
ways of learning
about teaching,
Journal of Curriculum Studies, vol. 23, pp. 393-395."
The author was
unable to find descriptions
of the five areas in the article.
(32)
OECD,
(33)
Dei,
(34)Ibid,
op. cit.,
2005,
p. 98
p. 283
p. 279
Reference
Ammermuller, A., 2005, Poor Background
or Low Returns?
Germany perform so poorly in PISA, ZEW DP 05-18.
175
Why immigrant
students
in
International
Cooperation
Project
Development
of Mathematics
Berry, J. W., Poortinga, Y. H., Segall, M. H. and Dasen, P. R., 1992,
Research and Applications,
Cambridge
University
Press.
Brown, G. and Micklewright,
J., 2003, Using TIMSS,
UNESCO Institute
for Statistics,
Montreal.
Dei,
G. J. S., 2005, The challenge
Comparative
Education,
Vol.41,
Cross-Cultural
PISA and IALS:
of inclusive
schooling
No.3, p.267-289.
Education
Psychology:
A View from the Outside,
in Africa:
a Ghanaian
case study,
Herskovits,
M. J., 1948, Man and His Works: the Science of Cultural
Anthropology,
Alfred A.
Knopf,NY.
Hofstede,
G. and Hofstede
G.J., 2005, Cultures
and Organizations:
Software of the Mind,
McGraw-Hill,
NY.
OECD, 2001, Knowledge and Skills for Life - First Results from PISA 2000, OECD, Paris.
OECD, 2004a,
Learning
for Tomorrow's World - First Results
from PISA 2003,
OECD, Paris.
OECD, 2004b, Reviews of National
Policies
for Education
- Denmark: Lessons from PISA
2000, OECD, Paris.
OECD, 2005, Teachers Matter: Attracting,
Developing
and Retaining
Effective
Teachers, OECD,
Paris.
Parker-Jenkins,
M., 1995, Children
of Islam: A Teacher's
Guide to Meeting
the Needs of
Muslim Pupils, Trentham Books, Staffordshire.
Pike,
K. 1967, Language in Relation
to a Unified Theory of the Structure
Janua Linguarum, Series Maior 24, The Hague: Mouton and Co..
Rangvid,
B. S., 2005,
Sources
of Immigrants'
Underachievement:
Copenhagen,
akf Working Paper, akf. Copenhagen.
Reagan, T. Non-Western Educational
Traditions:
Thought and Practice, 2005, LEA, London.
Red field, R., Linton, R., and Herskovits,
M., 1936,
American Anthropologist,
vol. 38, ppl49-152.
Samovar, L. and Porter,
R., 1991,
Communication
Indigenous
of Human Behavior,
Results
Approaches
Memorandum for the Study
between cultures,
Wadsworth
from
PISA-
to Educational
ofAcculturation,
Publishing
Secada, W.G. and Carey, D. A., 1990, Teaching
Mathematics
with Understanding
to Limited
English
Proficient
Students,
Urban Diversity
Series No. 101, ERIC Clearinghouse
on Urban
Education,
NY.
Schnepf,
S. V, 2004, How Different
Are Immigrants?
A Cross-Country
and Cross-Survey
Analysis
of Educational
Achievement,
IZA Discussion
Paper, Institute
for the Study of
Labor, Bonn.
176
Literature
Review
International
Cooperation Project Towards the Endogenous Development ofMathematics
Education
^
Final
Report
Literature
Prior Study of the Formation
Review
of Figural
Concepts
Mami Ishida
Hiroshima
University
The Formation of figural concept is very important in geometry. Because the main point of learning
geometry is that pupils acquire the ability of the abstract and the generalization
for their daily life.
In other words, through form the figural concepts, pupils can acquire them.
I describe
the study of the formation
Sort of the Figural
of figural
concepts
as follow.
Concepts
Matsuo (2000) told that mathematical
concept is one of general concept and needs abstract and
general thinking.
The mathematical concepts include many concepts such as figural concepts. Also
she divides the figural concepts into 3 parts as follow.
1.
object
concepts
This is the concept which can be gotten through
intheshape
2.
related
and generalization
of the figure
concepts
This is the concept which is relationship
3.
abstraction
between the figures.
any other concepts
Objected
concept
is very typical
concept
in geometry.
Because for example I draw a triangle
as
fol low.
As you can see, this is an equilateral
triangle. It means that it can not describe all triangles.
In other
words, if you try to explain any shapes, you can not do draw. You should use only the linguistic
methods. This is because object concept is very important and typical concept in geometry.
Formation
of the figural
concepts
Matsuo (2000) explained
that when the concept is formed, the concept image and the concept
definition
are used. The concept image means the sets of mental pictures which is relation to the
name of the concepts. The concepts definition
means the linguistic
formation which identifies
the
concepts. She explained
that when the concepts are formed, both linguistic
aspects and image
aspects is needed. According to above, we can understand that the concept definition
is common
concept and the concept image is individual
concepts.
177
International
Cooperation
Project
Education
Final
Report
Fischbein
(1993)
explained
that while image is a symbol of the sense, concept is a symbol of sign
and is used on the way of abstract thinking.
In addition,
the concept has universality.
It means that
the concept is almost same as the concept definition
which is mentioned by Matsuo (2000) and the
image is almost same as the concept image which is mentioned by Matsuo (2000).
Also Fischbein
(1993)
mentioned the relationship
between the concept definition
and the concept
image. According to him, these are in different
categories
basically.
It can be understood that the
concept definition
is very abstract and the concept image is very concrete. However he also told that
weshould not separate them but we should combine them in the situation which distills
the essence
from them. It means that through the relationship
between them, the concept is formed deeply.
In addition,
Fischbein
(1993)
mentioned that the image and the concept are related very much and
sometimes conflicted.
Also he told that the figural concepts are not growing automatically
and the
reason why geometry is very difficult
is that the figural concept can not be formed naturally.
In
terms of above, we can realize that the relationship
between the concept definition
and the concept
image is very important
when the figural concept is formed. Besides he mentioned that the
relationship
between the experiments which pupils have and the lessons which the pupils have is
very important and it should be intermittent.
The representation
system of the figural concepts
When pupils learn, the representation
is requisite.
Now I describe the sort of representation.
Nakahara (1 995) expressed five representations
in mathematics as follow.
1.
Realistic
Representation
This is included
the real world, the representation
experiments using the real or the embodiment figures.
2.
Manipulative
which
is used
the real,
the
Representation
This is the representation
which is occurred the concrete manipulation,
the concrete
real which is modeled or changed by somebody. It means the representation
which
exhibits the active manipulate in the teaching tools.
3.
Illustrative
Representation
4. Linguistic
diagrams
and graphs.
Representation
5.
which is used the pictures,
Symbolic Representation
In basic education
level,
representation
and linguistic
highest level. This is because
which is used local language.
which is used figures, words and arithmetic
operation
Realistic
representation,
manipulative
representation,
representation
are usually used. Symbolic representation
it is very far from using in basic education level.
Besides Realistic
representation,
manipulative
representation,
named the affinity of the representation
Linguistic
representation
named the rule of the representation.
178
and illustrative
and symbolic
illustrative
is in the
representation
representation
are
are
International
Cooperation
Project
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Final Report
The affinity of the representation
(Realistic
representation,
manipulative
representation,
and
illustrative
representation)
has the characteristic
which full of the concreteness, the imaginable
and
the intuition.
Because it includes
the meaning in the visual (image). That is why its characteristic
is
very intimate. In other hands, the rule of the representation
(Linguistic
representation
and symbolic
representation)has
the characteristic
which full of the strictness
and the objectiveness.
Because it
shows the detail. That is why its characteristic
lucks of the concreteness, the imaginable
and the
intuition.
By the mean of the above, the affinity of the representation
is relation
to the concept
image and the rule of the representation
is relation to the concept definition.
Nakahara(2001
), shougakusannsu3
osakashoseki
Manipulative
representatio
Linguistic
representatio
*
t:v.
J
. L?
-.representatio
å\
n
V:
Z*,K"»f
,-fc->'(,
i
2--i)!S
åA
4
A
/ 'A**
' «8,i.9ui
Z.i
/A
Realistic
representatio
n
The sense of the geometry
Here is explained the sense of the geometry. It means that how pupils used the figural concepts and
how pupils express. Kawasaki (2003) mentioned that the sense of the geometry is not only relation
to the physical
stimulates
which are gotten from the environment but also how the pupils think it or
how the pupils active. In addition,
he explained that the sense of the geometry can be divided into
two groups. The first is named the external sense, which is the geometrical
recognition
of the object
(things and shapes) of the out side. Also it can be divided into two groups. One it is the mental
manipulation
which recognizes the facts naked. It is called the intuition
of the visual. The other is
the mental manipulation
which recognizes
the intimate
meaning like the concepts and the
philosophy.
It is called the intuition
of the intimate which tends to identify the meaning of the facts.
The second is named the internal sense, which recognizes the immanent condition
of the objects
(things,
shapes) of the outside. Also it can be divided into two groups. One is the estimate of the
value to recognize the goodness, which estimates the figural information
and recognize on oriented
and intellect
from the external sense. The other is the sensibility,
which is the immanent recognition
of the sense or the feeling for the beauty of the figure.
179
International
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Final Report
External sense: the recognition
of the objects of the outside.
intuition
of the visual : the recognition
of the reality of the concrete things
Thetarget
isthereal
things •E •E - the real shapes
The recognition
of the figural element (shapes, sizes and positions)
in visual
•E intuition
of the intimate:
the recognition
of the object in the philosophy.
The target is the real things inthe philosophy
•E •E •E.general and universally
shapes
The recognition
of the figural elements and the relation linguistically
The intimate meaning of the shapes which are gotten by experiments or
learning,
(the individual
concepts )
Internal sense: the recognition
of the situation
of the objects (things, shapes) of the outside
•E estimate of the value:
the recognition
of the goodness
The recognition
of the estimation of the figural information and recognition
on oriented and intellect
from the external sense.
sensibility:
the recognition
of the beauty
The immanent recognition
of the beauty or the stability
of the shapes
Also Kawasaki (2001)
mentioned that the
external sense is estimated
and the internal
he told that it is the internal sense that the
beauty or the stability.
Kawasaki (200 1 ) shows the function of the
goodness of the information
which is identified
by the
sense is used for getting good information.
In addition
regulation
such as laying the tiles is recognized
as the
cognition
in geometrical
sense as follow.
( 1 ) recognition
of the plane
® part and hole
a. Gestalt •E •E •EWhen the shape is seen, the property or the structure of the shape is
realized as a whole part.
b. View and ground •E•E•EView is the part (shape) which is cognized the first. And ground
is the part (shape)
which tends to be missed so that the
impression is very shallow.
(D composition
and decomposition
c. structure of the cognition
•E •E •EThis is the composition
or the decomposition
of the
figures in cognition
d. transformation
of the cognition •E •E•ECongruent transformation(translation,
rotations
and symmetry) or similarity
( 2 ) recognition
of three dimension
e. homeostasis •E •E •E This is the manipulate
realization
of the figural
f. depth perception
•E •E •E This is the cognition
rawn in the plane.
180
which help the cognition
chunk in homeostasis.
that the depth is realized
through
the
by the figure
International
Cooperation
Education
Final Report
Reference
Fischbein
E (1993),
The Theory of Figural
Concepts,
Educational
Studies in
24.ppA39-l62.
Michihiro
Kawasaki
(2001),
zukeisidouniokeru
'zukeikannkaku'no
iminituite,
pp.93-103
Nanae
Mats uo
(2 0 03 ),
Shougakkous annsuni okeru
atarashi i zukeikyoui
Tottoridaigakukyouikukenkyu
(5) pp.293-3 05
Nanae Matsuo (2000),
sansu/sugakunikeru
zukeisidounokaizen,
touyoukannshuppan
S. Vinner (1983),
Concept definition,
concept image and the notion of function,
Education in Science and Technology 14(3).pp.293-305.
Tadao Nakahara (1995)
Study of the Representational
System in Mathematics
seibunnsha
181
Mathematics
JASME (7)
kunoari kata,
Mathematical
Education.
ANNEX
Research Tools
ANNEX: Research
1. Questionnaire
[General
for the fisrt
Tools
year survey (prototype)
Directions]
In this booklet,
you will find questions
questions ask for your opinions.
about yourself.
Some questions
Read each question carefully
and answer as accurately as possible.
not understand something or are not sure how to answer.
ask for facts
while
You may ask for help if you do
Some of the questions
will be followed by a few possible
choices indicated
with
these questions,
circle the answer of your choice as shown in Examples 1, 2, and 3.
a number. For
[Example 1]
Do you go to school?
Fil
in one circle only
Yes
/T)
No
2
[Example2]
Howoften do you do these things?
Fil
Every
day
a) I listen
to music
b) I talk with my friends
c) I play
sports
[Example3]
Indicate
in one circle
for each line
At least
Oncea
week
Once or
Twicea
month
A few
timesa
year
2
I
I3
(4^)
I
I5
1
-(T)-
3
4
5
2
-(3j)
4
how much you agree with each of these statements.
Fil in one circle for each line
Agree
Agree
Disagree
a lot
a little
a little
a) Watching
b) I like eating
Never
I1
1
movies is fun
1
ice cream
1
1
I
2
(^ )-
183
other
5
Disagree
a lot
fT\
I
4
1
3
4
Read each question carefully,
and pick the answer you think is best. There are no right or wrong
answers. Fill in the circle the number of your answer. If you decide to change your answer, erase
your first answer and then fill in the circle the number of new answer. Ask for help if you do not
understand something or are not sure how to answer.
Thank you for your time, effort,
and thought
in completing
this questionnaire.
[A b o u t Y o u ]
I.
W h e n w e re y o u b o rn ?
B . F ill in th e c ir c le n e x t to t h e m o n th y o u w e re
b o rn
A . F ill in th e c ir c le n e x t t o th e y e a r
M o n th
Y ea r
1 . 19 9 0
1 . Ja n u ary
2. 199 1
2 . F e b r u a ry
3. 1992
3 . M arc h
4 . 19 9 3
4 . A p r il
5 . 19 9 4
5 .M ay
6 . 19 9 5
6 . Ju n e
7 . 19 9 6
7 . J u ly
8 . 19 9 7
8 . A u g u st
9 . O th e r
9 . S e p te m b e r
10 . O cto b e r
1 1. N o v em b e r
1 2 . D e c em b e r
184
International
Cooperation
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Development
of Mathematics
Education
^
FinalReport
II.
Are you a girl or aboy?
Fil
Girl
Boy
in one circle
1
2
only
III.
Howoften do you speak <language
Always
Almost always
Sometimes
Never
oftest> at home?
Fill in one circle only
1
2
3
4
IV.
About how many books are there in your home? (Do not count magazines,
school books.)
Fill in one circle only
None or very few
(0-10 books)
1
Enough to fill one shelf
(1 1-25 books)
2
Enough to fill one bookcase
(26-1 00 books)
3
Enough to fill two or more bookcases
(more than 100 books)
4
V.
Do you have any of these items at your home?
Fill in one circle
a)
b)
c)
d)
e)
f)
g)
f)
Calculator
Television
Radio
Study desk/table
for your use
A quiet place to study
Dictionary
Books to help with your school work
Computer (do not include
TV/video game computers)
for each line
Yes
No
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
185
2
newspapers,
or your
International
Cooperation
Project
of Mathematics
Education
Final
[Mathematics
Report
in School]
VI.
Howmuch do you agree with these statements
about learning mathematics?
Agree
a lot
I
a) I usually do well in mathematics
b) I would like to do more mathematics
in school
c) Mathematics
is harder for me
than for many of my classmates
d) I enjoy learning mathematics
e) I amjust not good at mathematics
f) I learn things quickly
in mathematics
g) I think learning mathematics
will help me in my daily life
h) I need mathematics to
learn other school subjects
i) I need to do well in mathematics
to get into the university
of
my choice
j) I need to do well in mathematics to
get thejob I want
Agree
a little
Disagree
a little
1
2
I
1
1
2
3
4
1
1
1
1
2
2
2 2
3
3
-3
3
4
4
4
4
1
2
3
4
3
4
1
I
3
2
4
1
2
3
4
1
2
3
4
VII.
Howoften do you do these things
in your mathematics lessons?
Fil in one circle
Every or
almost
About
h alf the
everv
,
lesson
lessons
a) I practice adding, subtracting,
multiplying,
and dividing
without
using a calculator
b) I work on fractions and decimals
c) I measure things in the classroom
and around the school
d) I make tables, charts, or graphs
e) I learn about shapes such as circles,
triangles,
and rectangles
f) I relate what I am learning in
mathematics to my daily lives
g) I work with other students
in small groups
h) I explain my answers
-
for each line
c
borne
l esson s
esS0 S
Never
I
å¼
1
1
1
r
2
2
1
å¼
3
3
1
å¼
4
4
1
1
2
2
3
3
4
4
1
2
3 -
4
1
2
3
4
1
-1
-2
2
186
3
3
-4
4
International
Cooperation
Project
Education
Final Report
i) I listen to the teacher talk
j) I work problems on my own
k) I review my homework
1) I have aquiz ortest
m) I use a calculator
[Your
VIII.
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
School]
Howmuch do you agree with these statements
about your school?
Fill in one circle for each line
Agree
Agree
D isagree
a lot
a little
a little
a) I like being in school
b) I think that students in my school
try to do their best
c) I think that teachers in my school
care about the students
d) I think that teachers in my school
want students to do their best
1
1
1
2
3
1
1
1
2
3
4
1
2
3
4
1
2
3
IX.
In school,
a)
b)
c)
d)
did any of these things
happen during the last one year?
Yes
No
Something
of mine was stolen
1
2
I was hit or hurt by other student(s)
(for example, shoving, hitting, kicking)
1
2
I was made to do things I didn't
want to do by other students
1
2
I was left out of activities
by other
stud ents
1
2
187
4
4
International
Cooperation
Project
of Mathematics
Education
Final Report
[Things
You Do Outside
of School]
X.
On a normal school
these things?
day, how much time do you spend before
Fil
a)
b)
c)
d)
e)
f)
g)
No
time
Less
than
l hour
I
2
I watch television
I play or talk with friends
I dojobs athome
I play sports
I read a book for enjoyment
I use computer
I do homework
1
1
1
1
1
1
doing
for each line
More
than2
but less
1-2
than
hours
4 hours
each of
in one circle
1
1
or after school
I
1
3
2
2
2
2
2
2
4 or
More
hm]rs
3 3
3
3
3
3
I
4
-4
4
4
4
4
4
5
5
5
5
5
5
5
[More About You]
XI.
Including
yourself,
how many people live in your home?
Fil in one circle only
2-3 persons
2
4-6 persons
3
7-9 persons
4
10-12 persons
5
More than 12 persons
6
XII.
A. Does your mother (or stepmother
your town?
Fil in one circle only
B. Does your father (or stepfather
town?
Fil
in one circle
only
or female guardian)
Yes
No
1
2
or male guardian)
Yes
1
No
2
188
belong
belong
to <the major ethnic
to <the major ethnic
group> in
group> in your
International
Cooperation
Project
of Mathematics
Education
Final Report
2. Achievement
[Face
Sheet
tests
for the first
year survey (prototype)
for Part I]
Mathematics
Achievement
Part i
Test
a) This is a test on Mathematics.
The result which you achieve in this test will never affect your grade in school.
relaxed and never cheat.
Read the question carefully and answer what you think the best.
b) The time allocation
Feel
is 41 minutes.
c) Some questions are multiple-choice
type, and others short answer type.
All the questions have to be answered with clear writing.
d) Never use any calculator or multiplication
You can use a ruler, if necessary.
e) When the question
correct answer.
has some choices
table
for your calculation.
from (A) to (D) or (E), please
encircle
only one
f) When the question needs a correct short answer, please write down the answer in the
blank on a dotted line, or show your answer in an appropriate
place in the question.
g) You have to write your process to get the answer in a space of each question.
School
Name:
Class Name:
Sex (Please
Age:
circle
either):
Male or Female
years old
Your Name:
189
International
Cooperation
Project
of Mathematics
Education
Final
Report
[1-1]
Central School had a bottle collection.
Children
in each class brought empty bottles
The principal
made a bar graph of the number of bottles from five classes.
to school.
100
80
Number
of
Bottles
60
*;%M
ts :*s?*'i蝣
m
1
8
H
I SS :
TS If
II
40
:;&
m
II
Ot:*
1
f':": s:
sSmi
,* :
蝣K i
ｫW *!
&%
20
^^
I
M s.
B a r b e r 's
C la s s
Which
(A)
(B)
(C)
(D)
class
Miss
Mr.
Mrs.
Mr.
collected
M r.
C h y n 's
C la s s
^
H
J!
-*蝣
M s.
F r ie d m a n 's
C la s s
I
'i>;
蝣
M s.
G o n z a le z 's
C la s s
M r.
M a c k 's
C la s s
45 bottles?
Barber's class
Chyn's class
Friedman's
class
Mack's class
[1-2]
Jasmine made a stack of cubes of the same size. The stack had 5 layers
cubes. What is the volume of the stack?
and each layer had 10
(A)5cubes
(B) 15 cubes
(C) 30 cubes
(D) 50 cubes
[1-3]
Which shows 2/3 of the square shaded?
(A)
(B)
(C)
(E)
ｻ:'
i 'i * : : ! :
im m
m
; ;r .
i Si i -S F
190
International
Cooperation
Project
Education
Final
Report
it will
take
[1-4]
It takes Chris 4 minutes to wash a window. He wants to know how many minutes
him to wash 8 windows at this rate. He should
(A) multiply
4x8
(B) divide
8 by 4
(C) subtract 4 from 8
(D) add 8 and 4
[1-5]
Here is a calendar
for December.
D E C EM B E R
s
Mary's birthday
what date will
(A) December
(B) December
(C) December
(D) December
M
T
w
T
F
s
1
2
3
4
5
6
7
8
9
10
ll
12
13
14
15
16
17
18
19
20
2 1
22
23
24
25
26
27
28
29
30
3 1
is on Thursday, December 2. She is going
she go on the trip?
16th
2 1 st
23rd
30th
on a trip exactly
[1-6]
Which digit
is in the hundreds
place in 2345?
(A)2
(B)3
(C)4
(D)5
[1-7]
Which number would be rounded
(A)62
(B)160
(C)546
(D)586
(E)660
to 600 when rounded
191
to the nearest
hundred?
3 weeks later. On
International
Cooperation
Project
of Mathematics
Education
Final
I1"8!
Which ofthesemeans
(A) 70
(B) 7
(C) 0.7
(D) 0.07
L
^q
[1-9]
One centimeter
on the map represents
About how far apart are Oxford
(A) 4 km
(B) 16km
(C) 35 km
(D) 50 km
8 kilometers
and Smithville
on the land.
on the land?
[1-10]
Which of these
(A) 1 kg
(B)6kg
(C) 60 kg
(D) 600 kg
could
be the weight
(mass)
of an adult?
[1-11]
Which of
(A) Nine
(B) Nine
(C) Nine
(D) Nine
these is
thousand
thousand
thousand
hundred
a name for 9740?
seventy-four
seven hundred forty
seventy-four hundred
seventy-four thousand
192
Report
International
Cooperation
Project
of Mathematics
Education
Final Report
[1-12]
å¡ represents the number of magazines that Lina reads each week. Which
the total number of magazines that Lina reads in 6 weeks?
(A)
6+D
(B)
6xQ
(C)
n+6
(D)(D+D)x6
[2-1]
This table
shows the ages of the girls
A ge
8
9
10
and boys in a club.
N u m b e r o f G ir ls
4
8
6
Use the information
in the table to complete
N u m b er o f B o y s
6
4
10
the graph for ages 9 and 10.
S-H
10
[2-2]
i
Sam said
that
^
^
ofapieislessthan
*
ofthesamepie.
Is Sam correct?
Use the circles below to show why this is so.
1
Shade
in
of this
Shadein
circle.
193
a
ofthiscircle.
of these
represents
International
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Final
Report
[2-3]
These shapes are arranged
in a pattern.
OAOOAAOOOAAA
Which set of shape is arranged
(A)
*å¡*å¡**å¡å¡**å¡å¡
(B)
å¡*å¡å¡*å¡å¡å¡*å¡å¡å¡å¡
(C)
*å¡**å¡å¡***å¡å¡å¡
(D)
å¡å¡*•Eå¡•Eå¡å¡*•Eå¡*
in the same pattern?
[2-4]
1
There are 600 balls
in abox,
and
^
of the balls
are red. How many red balls
box?
Answer :
[2-5]
The graph shows the number of cartons of milk sold each day ofa week at a school.
40
2
t
/5
30
,0
20
io
4Mon.
Tues.
Wed.
Thurs.
Fri.
Day
How many cartons of milk did the school
Answer:
Howmany cartons of milk did the school
Answer:
sell on Monday?
sell
that week? Show your work.
194
are in the
International
Cooperation
Project
Education
Final Report
[2-6]
Which of these could equal 1 50 milliliters?
(A) The amount of water in a cup
(B) The length ofa kitten
(C) The weight of an egg
(D) The area ofa coin
[2-7]
Lia is practicing
369?
Answer:
addition
and subtraction
problems.
What number should
Lia add to 142 to get
[2-8]
T
L_
I
I
^
.
_i
.j
.
I
.
1
1
å
!Bt
r
f
i
f--
rá"
.._,!
!
.
Draw a triangle in the grid so that the line AB is the base of the triangle
are the same length as each other.
and the two new sides
[2-9]
Mr. Brown goes for a walk and returns to where he started
30 minutes, at what time did he start his walk?
Answer:
[2-10]
The graph shows 500 cedar trees and 150 hemlock
Cedar
Hemlock
How many trees
does
each
represent?
Answer :
195
trees.
at 07:00.
If his walk
took 1 hour and
International
Cooperation
Project
Development
of Mathematics
Education
Final
Report
[3-1]
Central School had a bottle collection.
Children
in each class brought empty bottles
The principal
made a bar graph of the number of bottles from five classes.
to school.
100
Number
of
B ottles
80
m
:WK
%
Si ｻ
; ; ( ?蝣;
60
40
20
r iiS fts
s蝣
im蝣蝣
I. .- . .
I
M
l^ ^ ^ ^ ^ ^ ^ ^ H fi
M s.
B a r b e r 's
C la s s
蝣
M r.
C h y n 's
C la s s
蝣
M s.
G o n z a le z 's
C la s s
M s.
F r ie d m a n 's
C la s s
M r.
M a c k 's
C la s s
Which two classes collected
exactly 80 bottles?
(A) Miss Barber's and Mrs. Friedman's classes
(B) Miss Barber's and Mr. Mack's classes
(C) Mrs. Friedman's
and Miss Gonzalez's classes
(D) Miss Gonzalez's
and Mr. Mack's classes
[3-2]
When Tracy left for school, the temperature
was minus 3 degrees.
10
r
lO
i
io"3
0
-
-10
10
r¥
-
10
-10
At recess, the temperature
Howmany degrees did the temperature
was 5 degrees.
(A)
(B)
(C)
(D)
196
2
3
5
8
degrees
degrees
degrees
degrees
rise?
International
Cooperation
Project
Education
Final Report
[3-3]
Figures
that are the same size and shape are called
1
W
(A
(B
(C
(D
2
h ich tw o fig u re s a re c o n g ru e n t?
) 1 and 2
) 1 and 3
) 1 and 4
) 3 an d 4
s[3uje
-4 ]ct:
4 ,0 3
- L 15
(A
(B
(Q
(D
(E
) 5 .18
) 4 .4 5
3 .12
) 2 .9 8
) 2 .8 8
[3-5]
In this diagram,
2 out of every 3 squares are shaded.
Which diagram
has 3 out of every 4 squares shaded?
(B)
(A)
r?i
(C)
:m
I
IS
W J m iM w
・
(5L
mm
9 ,
KH jtS K K K S S
S SH SSSSm W
197
congruent
figures.
International
Cooperation
Project
of Mathematics
Education
Final
Report
[3-6]
This chart shows temperature
reading
made at different
times on four days.
T E M PER A T U R E
6 a .m .
9 a .m .
N o on
3 p .m .
8 p .m .
M o nd ay
15C
17-
2 0-
2 1C
19-
T u e sd a y
15C
15C
15C
10-
9-
W e d n e sd a y
8-
10 -
14C
13C
15 -
T h u rsd a y
8-
l lc
14 -
17-
20 -
What was
(A) Noon
(B) 3 p.m.
(C) Noon
(D) 3 p.m.
the highest
temperature
on Monday
on Monday
on Tuesday
on Wednesday
recorded?
[3-7]
1
Janis,
Maija,
Maija
ate
(A)
-
and their
mother
were eating
a cake. Janis
1
ate
á" of the cake.
1.
7
of the cake. Their mother ate
(B)
4
I
(C)
2
I
^
ofthecake.
Howmuchofthecake
is left?
(D)None
4
[3-8]
What number equals
(A)
6453
(B)
60453
(C)
64530
(D)
354060
(E)
604530
3 ones+
5 tens+4
hundreds+
60 thousands?
[3-9]
What units would be best to use to measure the weight
(A) centimeters
(B) milliliters
(C) grams
(D) kilograms
(mass)
of an egg?
[3-10]
All of the pupils
shape is a triangle."
(A) The shape has
(B) The shape has
(C) The shape has
(D) The shape has
in a class cut out paper shapes. The teacher picked
Which of these statements MUST be correct?
three sides.
a right angle.
equal sides.
equal angles.
198
one out and said,
"This
International
Cooperation
Project
Development
of Mathematics
Education
Final Report
[3-11]
There are 9 boxes of pencils.
(A)1025
(B)1100
(C)1125
(D)1220
(E)1225
Each box has 125 pencils.
What is the total
number
of pencils?
[3-12]
Each small square (å¡) is equal to 1. There
small squares in each large square.
Då¡
D å¡
Då¡
are 10 small
蝣
!蝣
! ! 蝣I II
蝣
IM
Iii ]
蝣
ii
I i in
蝣I
n !蝣! i
ll
1-5 u44
蝣
蝣蝣
!蝣
;!蝣
!a!蝣I
i
S:
蝣
ii
!i i
iiiiiiIH1
i !!
i
I- j
II!
蝣!
蝣
!I蝣!;I
!蝣!
I !蝣!蝣
!!
I
,
I I
3 s ii i
i
ii
i 4Hi
蝣1
ii
Il l !蝣 I
蝣
Iil i i
I 蝣
ii i 蝣
i
n
M
l
蝣
I
[蝣I
R
What number is shown?
(A) 16
(B) 358
(C) 538
(D) 835
[3-13]
Which of these figures
has the largest
area?
(A)
(B)
(C)
(D)
ir
J,
199
[I]
squares
in each strip.
There
are 100
International
Cooperation
Project
Development
of Mathematics
Education
Final Report
[B ackshe et]
ThankYou
for Your Cooperation!
The research
is conducted
by the
Cooperation,
1-5-1 Kagamiyama, Higashi-Hiroshima,
Graduate
School
for
International
Development
and
Japan 739-8529
All the test items are from TIMSS 2003 or TIMSS
TIMSS 2003 Mathematics
Items Released Set for
Copyright
© 2003 International
Association
for
(IEA)
TIMSS Mathematics
Items: Released
Set for
Reproduced from TIMSS Population
1 Item Pool.
Copyright
© 1994 by IEA, The Hague
1995.
Fourth Grade.
the Evaluation
Population
of Educational
1 (Third
and
Achievement
Fourth
Grades),
[Face Sheet for Part II]
Mathematics
a) This
The
and
Read
Achievement
PartII
Test
is a test on Mathematics.
result which you achieve in this test will never affect your grade in school.
never cheat.
the question carefully
and answer what you think the best.
c) Some questions
All the questions
Feel relaxed
is 43 minutes.
are multiple-choice
type, and others short answer type.
have to be answered with clear writing.
d) Never use any calculator or multiplication
e) When the question
has some choices
table
for your calculation.
from (A) to (D)
or (E),
please
encircle
only
one correct
answer.
f) When the question needs a correct short answer, please write down the answer in the blank
dotted line, or show your answer in an appropriate
place in the question.
g) You have to write your process to get the answer in a space of each question.
School
Name:
Class Name:
Sex (Please
Age:
circle
either):
Male or Female
years old
Your Name:
200
on a
International
.
Cooperation
films should
a
a
a
a
Development
of Mathematics
Education
Final
[4-1]
Simon wantsto
(A)
(B)
(C)
(D)
Project
watch afilm
that is between
x
1å tj"
and 2 hours
long.
Which
Report
of the following
be chosen?
59-minutes
102-minutes
121-minutes
150-minute
film
film
film
film
[4-2]
What is the smallest
digit only once.
Answer:
whole
number that you can make using the digits
4, 3, 9 and 1? Use each
[4-3]
204-4 =
Answer :
[4-4]
Tanya has read the first 78 pages in a book that is 130 pages long. Which number sentence could
Tanya use to find the number of pages she must read to finish the book?
(A)
(B)
(C)
(D)
130+78=D
D-78=130
130-78=å¡
130-78=D
[4-5]
Here is a number sentence.
4xD<17
Which
number could go in the
å¡ to make the sentence true?
(A)4
(B)5
(C)12
(D)13
201
International
Cooperation
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Final
Report
[4-6]
On this grid, find the dot with the circle around it. We can describe
is a FirstNumber
1, Second Number 3.
where this dot is by saying
£
Jb,
§
o
0
1
2
3
4
First Number
Nowfind the dot with the triangle
around it. Describe where the dot is on the grid in the same
way. Fill in the numbers we would use:
First number
Second Number
[4-7]
A. Draw 1 straight
line on this rectangle
to divide
it into 2 triangles.
B. Draw 1 straight
line on this rectangle
to divide
it into 2 rectangles.
C. Draw 2 straight
lines
on this rectangle
to divide
[4-8]_
Additional
4+4+4+4+4
Write this addition
Fact
=20
fact as a multiplication
fact.
202
it into
1 rectangle
and 2 triangles.
it
International
Cooperation
Project
Education
Final Report
[4-9]
A teacher marks 10 of her pupils'
tests every half hour. It takes
all her pupils'
tests. How many pupils are in her class?
Answer:
her one and a half hours to mark
[4-10]
Ali had 50 apples.
shows this?
(A) å¡-20=50
(B) 20-D=50
(C) å¡-50=20
(D) 50-D=20
He sold some and then had 20 left. Which of these is a number sentence that
[5-1]
The graph shows the heights
offour
girls.
150
+-|
^125
100
fl
75
B
50
25
0
Name of Girls
The names are missing from the graph.
than Sarah. How tall is Sarah?
(A)75cm
(B)100cm
(C)125cm
(D)150cm
Debbie
is the tallest.
Amy is the shortest.
[5-2]
Here is a cone. Part of its surface
Which
( A)
of these solids
is flat and part of its surface
also has both a flat surface
( B)/
( C)
203
is curved.
and a curved surface?
(DL
Dawn is taller
International
Cooperation
Project
of Mathematics
Education
Final
Report
[5-3]
Mark's garden has 84 rows of cabbages. There are 57 cabbages in each row. Which
gives the BEST way to estimate how many cabbages there are altogether?
(A) 100x50=5000
(B) 90x60=5400
(C) 80x60=4800
(D) 80x50=4000
of these
[5-4]
Which of these has the same value as 342?
(A) 3000+400+2
(B) 300+40+2
(C) 30+4+2
(D)3+4+2
[5-5]
What isthe
(A) 5.3
(B) 6.3
(C) 6.4
(D) 9.5
sum of2.5
and 3.8?
[5-6]
In which figure are one-halfofthe
dots black?
o
o o
o * 'Oo
o
・
(A)
(B)
o
・
(D)
(C)
[5-7]
In Toshi's class there are twice as many girls as boys. There are 8 boys in the class. What is the
total number of boys and girls in the class?
(A)12
(B)16
(C)20
(D)24
204
International
Cooperation
Project
of Mathematics
Education
Final Report
[5-8]
This figure will be turned to a different
position.
N
Which of these could be the figure
( A).
( B)
after it is turned?
(C)
n
(Ph
$0.
a
[5-9]
Here is a rectangle
with length
its shape is called its perimeter.
6 centimeters
and width
4 centimeters.
6cm
4cm
Which of these gives
(A) 6+4
(B)6x4
(C) 6x4x2
(D)6+4+6+4
the perimeter
of the rectangle
[5-10]
Which number sentence
(A) 968<698
(B) 968<689
(C) 968>689
(D) 968=689
is true?
[5-11]
Here is a number pattern.
100,1,99,2,98,
What three numbers should
(A)3,
(B) 4,
(C) 97,
(D)97,
97,
97,
3,
4,
å¡ , å¡ , å¡
go in the boxes?
4
5
96
96
205
in centimeters?
The distance
right
around
International
Cooperation
Project
of Mathematics
Education
Final
Report
[5-12]
Which number is equal
(A)17
(B) 170
(C) 1700
(D) 17000
to eight
tens plus nine tens?
[6-1]
Which pair
number"?
of numbers
the rule
"Multiply
the first
number by 5 to get the second
3
(A)15(B)6
ll
'
6
(C)ll.
(D)
follows
15
3
[6-2]
Craig folded
a piece of paper in half and cut out a shape.
fold
Draw a picture to show what the cut-out shape will look like
out.
[6-3]
About how long is this picture
ofa pencil?
(A) 5 cm
(B) 10 cm
(C) 20 cm
(D) 30 cm
[6-4]
Which
(A) 1
(B) 1
(C) 1
(D) 1
of these
kilogram
centigram
milligram
gram
is largest?
206
when it is opened
up and flattened
International
Cooperation
Project
of Mathematics
Education
Final
[6-5]
What is 5 less than 203?
Answer :
[6-6]
Henry is older than Bill, and Bill is older than Peter. Which statement
(A) Henry is older than Peter.
(B) Henry is younger than Peter.
(C) Henry is the same age as Peter.
(D) We cannot tell who is oldest from the information.
[6-7]
The daily
S how
S ta r t T im e
1 st
2 :0 0 p .m .
2 nd
3 :3 0 p . m .
3 rd
5 :0 0 p . m .
4 th
?
If this
(A)
(B)
(C)
(D)
start times for showing
pattern
continues,
a movie are listed
below:
what is the start time for the 4th time?
5:30p.m.
6:00
p.m.
6:30
p.m.
7:00
p.m.
[6-8]
These numbers are part ofa pattern.
50,46,42,38,34,...
What do you have to do to get the next number?
Answer :
[6-9]
On the grid,
!
i.
iI
t
I
!
i
I
1!
I
i-
L
T
I
i
{
i!
i
rI
I
to line L.
f" - f
t
t
i
3
II
f
--
iI
draw a line parallel
i
I
I
1
!
・t
i
I
!
I
i
1
1
1
i
207
must be true?
Report
International
Cooperation
Project
Education
Final
Report
[6-10]
0
On the number line above, what number goes in the box?
Numberin
å¡ =
[6-11]
Howmany millimeters
Answer :
are in meter?
[6-12]
37x0=703
What is the value of37xD+6?
Answer :
[6-13]
Write
a fraction
that is larger
than
2
7
Answer :
[B ackshe et]
Thank You
for Your Cooperation!
The research
is conducted
by the
Cooperation,
1 -5- 1 Kagamiyama, Higashi-Hiroshima,
Graduate
School
for International
Development
and
Japan 739-8529
All the test items are from TIMSS 2003 orTIMSS
TIMSS 2003 Mathematics
Items Released Set for
Copyright
© 2003 International
Association
for
(IEA)
TIMSS Mathematics
Items: Released
Set for
Reproduced from TIMSS Population
1 Item Pool.
Copyright
© 1994 by IEA, The Hague
208
1995.
Fourth Grade.
the Evaluation
Population
of Educational
1 (Third
and
Achievement
Fourth
Grades),
International
Cooperation
Project
of Mathematics
Education
Final
3. Achievement
[Face
test for the second
Report
year survey (Prototype)
Sheet]
Mathematics
Achievement
Test 2005
a) The result which you achieve in this test will never affect your grade in school.
nevercheat.
Read the questions carefully
and answer what you think is the best.
is 40 minutes.
c) Some questions are multiple-choice
type, and others are short answer type.
All the questions have to be answered with clear writing.
d) Never use any calculator
or multiplication
table for your calculation.
e) You have to write your process to get the answer in the space of each question.
Your School
Feel relaxed
Name:
Your Grade:
Sex (Please
circle
Age:
either):
Male or Female
years old
Your Name:
Q1. Work out the following
1
0)
calculations.
2
T~+T"=
2
7
(2)
7
Q2. Express the answer in a fraction.
3-7=
Q3. The following
following
(Example)
figures
show lm of a bar. Shade
length.
1
4
m
^
4 m
lm
209
a part of the figure
to represent
the
and
International
Cooperation
Project
of Mathematics
Education
Final
Report
lm
(1)
m
lm
(2)
.m
Q4. The following
figures
following
length.
1
(1)
(2 )
show 2m of a bar. Shade
a part of the figure
to represent
the
2m
-o f 2 m
2m
.m
2
(3 )
1
Q5. Which
bracket.
1.
2m
蝣m
of the following
is greater
AisbiggerthanB.
than the other?
2.
.4
( 1)
A:
B
(2)
A:A_
B:A
(3)
A:
B:
:
(
Answer:
(
Answer:
(
_
2_
(1)
Whenyouadd
5
£
3.
)
answer.)
J_
ofwaterto
5
£ofwaterinacontainer,
how much water is in the container?
(2)
Whenyoucut
7
what is the remaining
a correct
BisbiggerthanA.
Answer:
Q6. Answer the following
questions.
(Show your process to get the correct
Write
mawayfrom
7
mofastring,
length?
210
choice
A=B
(1, 2 or 3) in the
International
Cooperation
Project
of Mathematics
Education
Final
(3)
When you arrange 6 pieces
what is the total
Report
of ~T~m
paper in a line,
length?
Q7. Change the decimal number " 0.7 " to a fraction.
(Show your process to get the answer.)
Q8. Judy bought 5kg of meat. George bought 3kg of meat. Write a fraction in the bracket
between Judy's meat and George's meat in terms of weight.
George's meat is (
) ofJudy's
meat.
Q9. Which of the following
represents
"half of circle(s)
is shaded.
L oo
00
3.
"half for you1? Circle
all the possible
2- oo
O
°o°
4.
o°o
_
i^lU. imagine
and make a sentence
._
problem in which the answer is
2
3
'
(You can express your sentence problem in words and/or any diagrams.)
Thank You Very Much for Your Cooperation!
The Graduate School for International
Development
1 -5- 1 Kagamiyama, Higashi-Hiroshima,
Japan 739-8529
211
and Cooperation,
choice(s)
in which
to
International
Cooperation
Project
ofMathematics
Education
Final Report
4. Interview
[Interview
items for the second year survey
Items for Pupils]
Select ten pupils on the basis of their latest end of term exam about mathematics.
They should consist of the top five pupils who usually get a good grade in mathematics
and the bottom five who are not good at mathematics.
Pupil's
Name:
Item 1: Inquire
what choice they selected
and why in Q5 (3).
i) Ask them to read to you the sentence ofQ5
YES
NO
i ) If there
is any difficult
i i) Ask them to explain
iv) Describe
any specific
term/expression
(3). Can they correctly
for them to read, describe
how they solve the question
mistakes,
if there are any.
212
read it?
it below.
in words and/or any diagrams.
International
Cooperation
Project Towards the Endogenous Development
of Mathematics
Education
Final
Item 2: Inquire
how they solved
Q6 (1).
i) Ask them to read to you the sentence
YES
NO
i ) If there
is any difficult
iv)
term/expression
ofQ6 (1).
Can they correctly
how they solved the question
Ask them to write their
read it?
below.
numerical
expression
to find the answer for Q6 (1).
v) Ask them to calculate
YES
the numerical
NO
expression.
Can they correctly
vi) Describe
mistakes
any specific
in the calculation,
213
if there
are any.
calculate
it?
Report
International
Cooperation
Project
Education
Final
Item 3: Inquire
how they solved
the Q8.
i) Ask them to read to you the sentence
YES
NO
i ) If there
is any difficult
iv) Describe
any specific
ofQ8.
term/expression
Can they correctly
how they solve the question
mistakes,
if there
are any.
214
read it?
it below.
Report

十 - 広島大学学術情報リポジトリ - Hiroshima University

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十 - 広島大学 学術情報リポジトリ - Hiroshima University

Transcription

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十 - 広島大学学術情報リポジトリ - Hiroshima University